Research ArticleLie Groupoids and Generalized Contact Manifolds

Fulya Fahin

Department of Mathematics Faculty of Science and Art Inonu University 44280 Malatya Turkey

Correspondence should be addressed to Fulya Sahin fulyasahininonuedutr

Received 4 December 2013 Revised 3 June 2014 Accepted 5 June 2014 Published 23 June 2014

Academic Editor Jaume Gine

Copyright copy 2014 Fulya Sahin This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We investigate relationships between Lie groupoids and generalized almost contact manifolds We first relate the notions ofintegrable Jacobi pairs and contact groupoids on generalized contact manifolds and then we show that there is a one to onecorrespondence between linear operators and multiplicative forms satisfying Hitchin pair Finally we find equivalent conditionsamong the integrability conditions of generalized almost contact manifolds the condition of compatibility of source and targetmaps of contact groupoids with contact form and generalized contact maps

1 Introduction

A groupoid is a small category in which all morphisms areinvertible More precisely a groupoid (119866 119866

0) consists of two

sets 119866 and 1198660 called arrows and objects respectively with

maps 119904 119905 119866 rarr 1198660called source and target It is equipped

with a composition119898 1198662

rarr 119866 defined on the subset 1198662=

(119892 ℎ) isin 119866 times 119866 | 119904(119892) = 119905(ℎ) (see Figure 1) an inclusionmap of objects 119890 119866

0rarr 119866 and an inverse map 119894 119866 rarr 119866

For a groupoid the following properties are satisfied 119904(119892ℎ) =119904(ℎ) 119905(119892ℎ) = 119905(119892) 119904(119892minus1) = 119905(119892) 119905(119892minus1) = 119904(119892) and 119892(ℎ119891) =

(119892ℎ)119891 whenever both sides are defined 119892minus1119892 = 1119904(119892)

119892119892minus1 =1119905(119892)

Here we have used 119892ℎ 1119909and 119892minus1 instead of 119898(119892 ℎ)

119890(119909) and 119894(119892) Generally a groupoid (119866 1198660) is denoted by the

set of arrows119866 A topological groupoid is a groupoid119866whoseset of arrows and set of objects are both topological spaceswhose structuremaps 119904 119905 119890 119894 119898 are all continuous and 119904 119905 areopen maps

A Lie groupoid is a groupoid 119866 whose set of arrowsand set of objects are both manifolds whose structure maps119904 119905 119890 119894 119898 are all smooth maps and 119904 119905 are submersions Thelatter condition ensures that 119904 and 119905-fibres are manifoldsOne can see from the above definition that the space 119866

2of

composable arrows is a submanifold of 119866 times 119866 We note thatthe notion of Lie groupoids was introduced by Ehresmann[1] Relations among Lie groupoids Lie algebroids and otheralgebraic structures have been investigated by many authors[2ndash6]

On the other hand Lie algebroids were first introducedby Pradines [7] as infinitesimal objects associated with theLie groupoids More precisely a Lie algebroid structure on areal vector bundle 119860 on a manifold 119872 is defined by a vectorbundle map 120588

119860 119860 rarr 119879119872 the anchor of 119860 and an R-Lie

algebra bracket on Γ(119860) [ ]119860satisfying the Leibnitz rule

[120572 119891120573]119860= 119891[120572 120573]

119860+ 119871120588119860(120572)

(119891) 120573 (1)

for all 120572 120573 isin Γ(119860) 119891 isin 119862infin(119872) where 119871120588119860(120572)

is the Liederivative with respect to the vector field 120588

119860(120572) where Γ(119860)

denotes the set of sections in 119860In [8] Hitchin introduced the notion of generalized

complex manifolds by unifying and extending the usualnotions of complex and symplectic manifolds Later suchmanifolds have been studied widely by Gualtieri He alsointroduced the notion of generalized Kahler manifold [9]On the other hand the concept of generalized almost contactstructure on odd-dimensional manifolds has been studied in[10ndash12]

Recently Crainic [13] showed that there is a close rela-tionship between the equations of a generalized complexmanifold and a Lie groupoidMore precisely he obtained thatthe complicated equations of suchmanifolds turn into simplestructures for Lie groupoids

In this paper we investigate relationships between thecomplicated equations of generalized contact structures andLie groupoids We showed that the equations of such man-ifolds are useful to obtain equivalent results on a contact

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 270715 8 pageshttpdxdoiorg1011552014270715

2 Abstract and Applied Analysis

m(g h)

g h

G

G0

t(m(g h)) = t(g) s(g) = t(h) s(h) = s(m(g h))

Figure 1 Two arrows 119892 and ℎ isin 119866 with the target of ℎ 119905(ℎ) isin 1198660

equal to the source of 119892 119904(119892) isin 1198660 and the composed arrow119898(119892 ℎ)

[5]

groupoid The paper is organized as follows In Section 2 wegathermain definitions and results used in the other sectionsIn Section 3 we first state necessary and sufficient conditionsfor generalized almost contact structure to be integrable andthen we obtain a relation between integrable Jacobi pairs andcontact groupoids defined on generalized manifolds More-over we observe that there is a close relationship between(1 1)-tensors satisfying certain conditions in terms of tensorfields defined on generalized manifolds and multiplicativeforms Finally we find one to one correspondence amonggeneralized contact map source and target maps and theconditions of a generalized contact structure to be integrable

2 Preliminaries

In this section we give basic facts of Jacobi geometry Liegroupoids and Lie algebroids We first recall notions ofcontact manifold and contact groupoid from [5] A contactmanifold is a smooth (odd-dimensional) manifold119872 with 1-form 120578 isin Ω

1(119872) such that 120578and(119889120578)119899

= 0 120578 is called the contactform of119872 Let119866 be a Lie groupoid on119872 and 120578 a form on Liegroupoid 119866 then 120578 is called 119903-multiplicative if

119898lowast120578 = 119901119903

lowast

2(119890minus119903) 119901119903lowast

1120578 + 119901119903

lowast

2120578 (2)

where119901119903119894 119866times119866 rarr 119866 119894 = 1 2 are the canonical projections

and 119903 119866 rarr R 119903(119892ℎ) = 119903(119892) + 119903(ℎ) is a function [14]A contact groupoid over a manifold 119872 is a Lie groupoid 119866

over 119872 together with a contact form 120578 on 119866 such that 120578 is 119903-multiplicative We recall that multiplicative of a 2-form 120596 isdefined by

119898lowast120596 = 119901119903

lowast

1120596 + 119901119903

lowast

2120596 (3)

We now recall the notion of Jacobi manifolds A Jacobimanifold is a smooth manifold 119872 equipped with a bivectorfield 120587 and a vector field 119864 such that

[120587 120587] = minus2119864 and 120587 [119864 120587] = 0 (4)

where [ ] denotes the Schouten bracket In this case (120587 119864)defines a bracket on119862infin(119872R) which is called Jacobi bracketand is given for all 119891 119892 isin 119862infin(119872R) by

119891 119892 = 120587 (119889119891 119889119892) + 119891119864 (119892) minus 119892119864 (119891) (5)

The Jacobi bracket endows119862infin(119872R)with a local Lie algebrastructure in the sense of Kirillov [15]

We now give a relation between Lie algebroid and Liegroupoid more details can be found in [16] Given a Liegroupoid 119866 on 119872 the associated Lie algebroid 119860 = Lie(119866)

has fibres 119860119909= Ker(119889119904)

119909= 119879119909(119866(minus 119909)) for any 119909 isin 119872 Any

120572 isin Γ(119860) extends to a unique right-invariant vector field on119866 which will be denoted by the same letter 120572 The usual Liebracket on vector fields induces the bracket on Γ(119860) and theanchor is given by 120588 = 119889119905 119860 rarr 119879119872

Given a Lie algebroid 119860 an integration of 119860 is a Liegroupoid 119866 together with an isomorphism 119860 cong Lie(119866) Ifsuch a 119866 exists then it is said that 119860 is integrable In contrastwith the case of Lie algebras not every Lie algebroid admitsan integration However if a Lie algebroid is integrable thenthere exists a canonical source-simply connected integration119866 and any other source-simply connected integration issmoothly isomorphic to 119866 From now on we assume that allLie groupoids are source-simply connected

We now recall the notion of 119868119872 form (infinitesimalmultiplicative form) on a Lie algebroid [17] which will beuseful whenwedealwith relations between Lie groupoids andLie algebroids An 119868119872 form on a Lie algebroid 119860 is a bundlemap

119906 119860 997888rarr 119879lowast119872 (6)

satisfying the following properties

(i) ⟨119906(120572) 120588(120573)⟩ = minus⟨119906(120573) 120588(120572)⟩

(ii) 119906([120572 120573]) = L120572(119906(120573)) minusL

120573(119906(120572)) + 119889⟨119906(120572) 120588(120573)⟩

for 120572 120573 isin Γ(119860) where 120588 = 120588119860and ⟨ ⟩ denotes the usual

pairing between a vector space and its dualIf 119860 is a Lie algebroid of a Lie groupoid 119866 then a closed

multiplicative 2-form 120596 on 119866 induces an 119868119872 form 119906120596of 119860 by

⟨119906120596(120572) 119883⟩ = 120596 (120572119883) (7)

For the relationship between 119868119872 form and closed 2-form wehave the following

Theorem 1 (see [17]) If 119860 is an integrable Lie algebroid and if119866 is its integration then120596 997891rarr 119906

120596is a one to one correspondence

between closed multiplicative 2-forms on119866 and IM forms of119860

Finally in this section we give brief information on thenotion of generalized geometry details can be found in [9] Acentral idea in generalized geometry is that119879119872oplus119879

lowast119872 shouldbe thought of as a generalized tangent bundle to manifold119872If 119883 and 120585 denote a vector field and a dual vector field on119872 respectively then we write (119883 120585) (or 119883 + 120585) as a typicalelement of 119879119872 oplus 119879

lowast119872 The Courant bracket of two sections(119883 120585) (119884 120578) of 119879119872 oplus 119879lowast119872 = TM is defined by

⟦(119883 120585) (119884 120578)⟧ = [119883 119884] +L119883120578 minusL

119884120585 minus

1

2119889 (119894119883120578 minus 119894119884120585)

(8)

where119889L119883 and 119894

119883denote exterior derivative Lie derivative

and interior derivative with respect to 119883 respectively TheCourant bracket is antisymmetric but it does not satisfy theJacobi identity Here we use the notations 120573(120587♯120572) = 120587(120572 120573)

Abstract and Applied Analysis 3

and 120596♯(119883)(119884) = 120596(119883 119884) which are defined as 120587♯ 119879lowast119872 rarr

119879119872120596♯ 119879119872 rarr 119879lowast119872 for any 1-forms120572 and120573 2-form120596 and

bivector field 120587 and vector fields119883 and 119884 Also we denote by[ ]120587the bracket on the space of 1-forms on119872 defined by

[120572 120573]120587= L120587♯120572120573 minusL

120587♯120573120572 minus 119889120587 (120572 120573) (9)

3 Lie Groupoids and GeneralizedContact Structures

In this section we first give a characterization for generalizedcontact structures to be integrable then we obtain certainrelationships between generalized contact manifolds andcontact groupoids We recall a generalized almost contactpair and then a generalized almost contact structure

Definition 2 (see [12]) A generalized almost contact pair(I 119865+120578) on a smooth odd-dimensionalmanifold119872 consistsof a bundle endomorphism I of 119879119872 oplus 119879

lowast119872 and a section119865 + 120578 ofTM such that

I +Ilowast= 0 120578 (119865) = 1

I (119865) = 0 I (120578) = 0 I2= minus119868119889 + 119865 ⊙ 120578

(10)

where119865⊙120578(119883+120572) = 120578(119883)119865+120572(119865)120578 for any119883+120572 isin Γ(TM)SinceI has a matrix form

I = [120593 120587♯

120579♯ minus120593lowast] (11)

where 120593 is a (1 1)-tensor 120587 is a bivector field 120579 is a 2-formand 120593lowast 119879lowast119872 rarr 119879lowast119872 is dual of 120593 one sees that ageneralized almost contact pair is equivalent to a quintuplet(119865 120578 120587 120579 120593) where 119865 is a vector field 120578 a 1-form

Definition 3 (see [12]) A generalized almost contact structureon119872 is an equivalent class of such pairs (I 119865 + 120578)

We now present two examples of generalized almostcontact manifolds

Example 4 (see [11]) An (2119899 + 1)-dimensional smoothmanifold 119872 has an almost contact structure (120593 119865 120578) if itadmits a tensor field 120593 of type (1 1) a vector field 119865 and a1-form 120578 satisfying the following compatibility conditions

120593 (119865) = 0 120578 ∘ 120593 = 0

120578 (119865) = 1 1205932= minus119894119889 + 120578 otimes 119865

(12)

Associated with any almost contact structure we have analmost generalized contact structure by setting

I = [120593 0

0 minus120593lowast] (13)

Example 5 (see [11]) On the three-dimensional Heisenberggroup 119867

3 we choose a basis 119883

1 1198832 1198833 and let 120572

1 1205722 1205723

be a dual frame For 119905 = 119903119888+ 119894119903119904 where 119888 = cos V and 119904 = sin Vfor some real number V we define

120593119905=

2119903119888

1 minus 1199032(1198832otimes 1205722+ 1198833otimes 1205723)

120590119905=

1199032 minus 2119903119904 + 1

1 minus 11990321205722and 1205723

120587119905=

1199032 + 2119903119904 + 1

1 minus 11990321198832and 1198833

120578 = 1205721 119865 = 119883

1minus 1198871198832+ 1198861198833

(14)

for any real numbers 119886 119887 We also define

I = [120593119905

120587♯

119905

120590119905

minus120593lowast119905

] (15)

Then J119905= (119865 120578 120601

119905 120587119905120590119905) is a family of generalized almost

contact structures

Given a generalized almost contact pair (I 119865 + 120578) wedefine

119864(10)

= 119890 minus 119894I (119890) | 119890 isin ker 120578 oplus ker119865

119864(01)

= 119890 + 119894I (119890) | 119890 isin ker 120578 oplus ker119865 (16)

The endomorphism I is linearly extended to the complexi-fied bundleTM otimesC It has three eigenvalues namely 120582 = 0120582 = 119894 = radicminus1 and 120582 = minus119894 The corresponding eigenbundlesare 119871119865oplus119871120578119864(10) and119864(01) where 119871

119865and 119871

120578are the complex

vector bundles of rank 1 generated with 119865 and 120578 respectivelyDefine

119871 = 119871119865oplus 119864(10)

119871 = 119871119865oplus 119864(01)

119871lowast= 119871120578oplus 119864(01)

119871lowast

= 119871120578oplus 119864(10)

(17)

Definition 6 (see [11]) Consider a generalized almost contactpair and let 119871 be its associated maximal isotropic subbundleOne says that the generalized almost contact pair is integrableif the space Γ(119871) of sections of 119871 is closed under the Courantbracket that is ⟦Γ(119871) Γ(119871)⟧ sub Γ(119871) In this case thegeneralized almost contact pair is simply called a generalizedcontact pair A generalized contact structure is an equivalenceclass of generalized contact pairs

In the sequel we give necessary and sufficient conditionsfor a generalized almost contact structure to be integrable interms of the above tensor fields We note that the followingresult was stated in [12]

Theorem 7 A generalized almost contact pair correspondingto the quintuplet (119865 120578 120587 120579 120593) is integrable if and only if thefollowing relations are satisfied

(C1)

(a) 1

2[120587 120587] = 119865 and (120587

♯otimes 120587♯) 119889120578

(b) [119865 120587] = minus119865 and 120587♯L119865120578

(18)

4 Abstract and Applied Analysis

(C2)

120593120587♯= 120587♯120593lowast

120593lowast[120572 120573]

120587= L120587♯120572120593lowast120573 minusL

120587♯120573120593lowast120572 minus 119889120587 (120593

lowast120572 120573)

(19)

(C3)

1205932+ 120587♯120579♯= minus119868119889 + 119865 ⊙ 120578

119873120593 (119883 119884) + 119889120578 (120593119883 120593119884) 119865 = 120587

♯(119894119883and119884

119889120579)

(20)

(C4)

120593lowast120579♯= 120579♯120593

119889120579120593 (119883 119884 119885)

= 119889120579 (120593119883 119884 119885) + 119889120579 (119883 120593119884 119885) + 119889120579 (119883 119884 120593119885)

(21)

(C5)

L119865120593 = 0 L

119865120579 = 0 (22)

where

[120572 120573]120587= L120587♯120572120573 minusL

120587♯120573120572 minus 119889120587 (120572 120573)

120579120593 (119883 119884) = 120579 (120593119883 119884)

(23)

We note that if (11) is a generalized contact structure then

I = [120593 minus120587♯

minus120579♯

minus120593lowast] (24)

is also a generalized contact structureI is called the oppositeofI In this paper we denote a generalized contact manifoldendowed withI by119872

As an analogue of aHitchin pair on a generalized complexmanifold a Hitchin pair on a generalized almost contactmanifold 119872 is a pair (119889120578 120593) consisting of a contact form120578 and a (1 1)-tensor 120593 with the property that 119889120578 and 120593

commute (ie 119889120578(119883 120593119884) = 119889120578(120593119883 119884)) We note that sincea generalized almost contact structure is equivalent to ageneralized almost complex structure on119872timesR the bivectorfield 120587 of the generalized almost contact structure is notnondegenerate in general But we emphasize that we areputting this condition for restricted case

Lemma 8 Let119872 be a generalized almost contact manifold If120587 is a nondegenerate bivector field on 119879119872

lowast-119878119901119886119899120578 119889120578 is theinverse 2-form (defined by (119889120578)

♯= (120587♯)

minus1) and120587 satisfies (20)then 120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894

119865119889120578) if and only if 119889120578(1205932119883119884) =

120593lowast119889120578(119883 119884)

Proof For 119883 isin 120594(119872) we apply (119889120578)♯to (20) and using the

dual structure 120593lowast we have

(119889120578)♯1205932(119883)

= minus(119889120578)♯(119883) minus (119889120578)

♯(120587♯120579♯ (119883)) + (119889120578)

♯(120578 (119883) 119865)

119889120578 (1205932119883119884) = minus119889120578 (119883 119884) minus 120579

♯(119883) (119884) + 119889120578 (120578 (119883) 119865 119884)

(25)

We obtain

120593lowast119889120578 (119883 119884) = minus119889120578 (119883 119884) minus 120579 (119883 119884) + 120578 (119883) 119889120578 (119865 119884)

(26)

Since (26) holds for all119883 and 119884 we get

120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894

119865119889120578) (27)

In a similar way one can get the converse

From now on when we mention a nondegenerate bivec-tor field 120587 we mean it is nondegenerate on 119879119872lowast-Span120578We note that if 119889120578 is the inverse 2-form of 120587 nondegenerate120587 on 119879119872

lowast-Span120578 implies that 119889120578 is also nondegenerate on119879119872-Span119865

We say that 2-form 120579 is the twist of Hitchin pair (119889120578 120593)Note that in this case 120593 is neither an almost contact structurenor torsion(119873

120593) free

Lemma 9 Let (119872 120578 120593 119865) be an almost contact manifold 119889120578and 120593 commute if and only if 119889120578 + 120593lowast119889120578 = 120578 and (119894

119865119889120578)

Proof We will only prove the sufficient condition We have

120593lowast119889120578 (119883 119884) + 119889120578 (119883 119884) minus 120578 and (119894

119865119889120578) (119883 119884) = 0 (28)

Since 120593lowast is dual contact structure we get

119889120578 (120593119883 120593119884) + 119889120578 (119883 119884) minus 120578 (119883) (119894119865119889120578) (119884) = 0 (29)

Substituting 120593119883 by119883 and using contact structure property

119889120578 (minus119883 + 120578 (119883) 119865 120593119884) + 119889120578 (120593119883 119884) = 0 (30)

Hence we obtain

minus119889120578 (119883 120593119884) + 119889120578 (120593119883 119884) = 0 (31)

which shows that 119889120578 and 120593 commute The converse is clear

Next we see that (C1) is satisfied automatically when onechooses 119889120578 as the 2-form which is the inverse of 120587 defined by(119889120578)♯= (120587♯)

minus1

Lemma 10 Let120587 be a nondegenerate bivector on a generalizedalmost contact manifold119872 and 119889120578 the inverse 2-form (definedby (119889120578)

♯= (120587♯)

minus1) Then 120587 satisfies (C1)

Proof Since 119889120578 is a closed form it is obvious due to [13Lemma 27]

Abstract and Applied Analysis 5

Definition 11 (see [14]) The Lie algebroid of the Jacobimanifold (119872 120587 119865) is 119879lowast119872 oplus R with the anchor 120588 119879lowast119872 oplus

R rarr 119879119872 given by

120588 (120596 120582) = (120587 119865)♯(120596 120582) = 120587

♯(120596) + 120582119865 (32)

and the bracket

[(120596 0) (120578 0)] = ([120596 120578]120587 0) minus (119894

119865(120596 and 120578) 120587 (120596 120578))

[(0 1) (120596 0)] = (L(119865)

120596 0) (33)

The associated groupoid

Σ (119872) = 119866 (119879lowast119872 oplusR) (34)

is called contact groupoid of the Jacobi manifold 119872 We saythat 119872 is integrable as a Jacobi manifold if the associatedalgebroid 119879lowast119872 oplus R is integrable (or equivalently if Σ(119872) issmooth)

Thus we have the following result which shows that thereis a close relationship between the condition (C1) and acontact groupoid

Theorem 12 Let 119872 be a generalized almost contact manifoldand 120578 a contact form There is a 1-1 correspondence between

(i) integrable Jacobi pair (119865 120587) on 119872 (ie (119865 120587) issatisfying (C1) integrable)

(ii) contact groupoids (Σ 120578) over119872

Proof Since (120578 119889120578) is a contact pair and (119865 120587) satisfies(C1) then (119865 120587) is an integrable Jacobi pair [18] FromDefinition 11 one sees that a contact groupoid is obtainedfrom an integrable Jacobi pair

The converse is clear

We now give the conditions for (C2) in terms of 119889120578 and120593

Lemma 13 Let 119872 be a generalized almost contact manifoldand 119889120578 a 2-form Given a nondegenerate bivector 120587 on119879119872lowast-120578 (ie 120587♯ = ((119889120578)

♯)minus1) and a map 120593 119879119872 rarr 119879119872

then 120587 and 120593 satisfy (C2) if and only if 119889120578 and 120593 commute

We now give a correspondence between generalizedcontact structures with nondegenerate 120587 and Hitchin pairs(119889120578 120593)

Proposition 14 There is a one to one correspondence betweengeneralized contact structures given by (11) with 120587 nonde-generate and Hitchin pairs (119889120578 120593) such that 119889120578(119883 119884) =

119889120578(120593119883 120593119884) In this correspondence 120587 is the inverse of 119889120578 and120579 is the twist of the Hitchin pair (119889120578 120593)

Proof Since (119889120578 120593) is Hitchin pair 119889120578 and (119889120578)120593are closed

By using the following equation (see [13])

119894119873120593(119883119884)

(119889120578) = 119894120593119883and119884+119883and120593119884

(119889(119889120578)120593) minus 119894120593119883and120593119884

(119889 (119889120578))

minus 119894119883and119884

(119889 (120593lowast119889120578))

(35)

we get

119894119873120593(119883119884)

(119889120578) = minus119894119883and119884

(119889 (120593lowast119889120578)) (36)

Since 120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894119865119889120578) we derive

(119889120578)♯(119873120593(119883 119884)) = minus119894

119883and119884(119889 (minus120579 + 120578 and (119894

119865119889120578))) (37)

Applying 120587♯ to (37) then we get

119873120593(119883 119884) = minus120587

♯(119894119883and119884

(119889 (minus120579 + 120578 and (119894119865119889120578))))

119873120593(119883 119884) = 120587

♯(119894119883and119884

(119889120579) minus 120587♯(119894119883and119884

119889120578 and (119894119865119889120578)))

119873120593(119883 119884) = 120587

♯(119894(119883and119884)

119889120579) minus 119889120578 (119883 119884) 119865

(38)

By assumption we have

119889120578 (119883 119884) = 119889120578 (120593119883 120593119884) (39)

Putting this equation into (38) we obtain

119873120593(119883 119884) = 120587

♯(119894(119883and119884)

119889120579) minus 119889120578 (120593119883 120593119884) 119865 (40)

which is the second equation of (C3) Now we show that120593lowast120579♯= 120579♯120593 From (26) we obtain

120593lowast120579♯= 120593lowast(minus(119889120578)

♯minus (120593lowast(119889120578))♯+ (120578 and (119894

119865(119889120578)))

♯) (41)

Hence we have

120593lowast120579♯= minus(119889120578)

♯120593 minus (120593

lowast(119889120578))♯120593 + (120578 and (119894

119865(119889120578)))

♯120593 (42)

From definition of twist we get

120593lowast120579♯= 120579♯120593 (43)

This equation is the first equation of (C4) Now wewill obtain

119889120579120593(119883 119884 119885)

= 119889120579 (120593119883 119884 119885) + 119889120579 (119883 120593119884 119885) + 119889120579 (119883 119884 120593119885)

(44)

which is second equation of (C4) Writing the equation as

119894119883and119884

(119889120579120593) = 119894120593119883and119884+119883and120593119884 (119889120579) + 120593

lowast(119894119883and119884 (119889120579)) (45)

and since 120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894

119865119889120578) then we should find

119894119883and119884

(119889 (minus(120593lowast(119889120578))120593+ (120578 and (119894

119865(119889120578)))

120593))

= 119894120593119883and119884+119883and120593119884

(119889 (minus120593lowast119889120578 + 120578 and (119894

119865119889120578)))

+ 120593lowast(119894119883and119884

(119889 (minus120593lowast119889120578 + 120578 and (119894

119865119889120578))))

(46)

A straightforward computation shows that

119894119883and119884

(119889 ((120593lowast(119889120578))120593)) = 119894

120593119883and119884+119883and120593119884(119889 (120593lowast119889120578))

+ 120593lowast(119894119883and119884

(119889 (120593lowast119889120578)))

(47)

6 Abstract and Applied Analysis

Using (35) then we get

119894119883and119884

(119889 ((120593lowast(119889120578))120593))

= 119894120593119883and119884+119883and120593119884

(119889 (120593lowast119889120578)) minus 119894

119873120593(119883119884)(119889120578)120593

(48)

Since 119894119873(119883119884)

((119889120578)120593) = 120593lowast119894

119873(119883119884)(119889120578) applying 119894

119883and119884119889(120593lowast119889120578)

= minus119894119873120593(119883119884)

(119889120578) to (48) we have

119894119883and119884

(119889 ((120593lowast(119889120578))120593))

= 119894120593119883and119884+119883and120593119884

(119889 (120593lowast119889120578)) + 120593

lowast(119894119883and119884

(119889 (120593lowast119889120578)))

(49)

In a similar way we can obtain (C5)The converse is clear from Lemmas 10 and 13

We note that similar to 2-forms given a Lie groupoid 119866a (1 1)-tensor 119869 119879119866 rarr 119879119866 is called multiplicative [13] iffor any (119892 ℎ) isin 119866 times 119866 and any V

119892isin 119879119892119866 119908ℎisin 119879ℎ119866 such that

(V119892 119908ℎ) is tangent to 119866 times 119866 at (119892 ℎ) so is (119869V

119892 119869119908ℎ) and

(119889119898)119892ℎ

(119869V119892 119869119908ℎ) = 119869 ((119889119898)

119892ℎ(V119892 119908ℎ)) (50)

Let (119872 120578) be a contact manifold Then it is easy to see thatthere is a one to one correspondence between (1 1)-tensors120593 commuting with 119889120578 and 2-forms on119872 On the other handit is easy to see that (C2) is equivalent to the fact that 120593lowast ∘ 119901119903

1

is an 119868119872 form on the Lie algebroid119879lowast119872oplusR associated Jacobistructure (119865 120587) Thus from the above discussion Lemma 13andTheorem 1 one can concludewith the following theorem

Theorem 15 Let119872 be a generalized almost contact manifoldLet (119865 120587) be an integrable Jacobi structure on 119872 and (Σ 120578)

a contact groupoid over 119872 Then there is a natural 1-1correspondence between

(i) (1 1)-tensors 120593 on119872 satisfying (C2)

(ii) multiplicative (1 1)-tensors 119868 on Σ with the propertythat (119868 119889120578) is a Hitchin pair

We recall the notion of generalized contact map betweengeneralized contact manifolds This notion is similar to thegeneralized holomorphic map given in [13]

Let (119872119894I119894) 119894 = 1 2 be two generalized contact

manifolds and let 120593119894 120587119894 120579119894be the components of I

119894in the

matrix representation (11) A map 119891 1198721

rarr 1198722is called

generalized contact if and only if 119891 maps 1205931into 120593

2 1198651into

1198652 and 120587

1into 120587

2 119891lowast1205792= 1205791and (119889119891) ∘ 120593

1= 1205932∘ (119889119891)

We now state and prove themain result of this paperThisresult gives equivalent assertions between the condition (C3)twist 120579 of (119889120578 119868) and contact maps for a contact groupoidover119872

Theorem 16 Let 119872 be a generalized almost contact manifoldand (Σ 120578 119868) an induced contact groupoid over 119872 with theinduced multiplicative (1 1)-tensor Assume that ((119865 120587) 119868)

satisfy (C1) (C2) with integrable (119865 120587) Then for a 120579 2-formon119872 the following assertions are equivalent

(i) (C3) is satisfied(ii) 119889120578 + 119868

lowast119889120578 minus 120578 and (119894119865119889120578) = 119904lowast120579 minus 119905lowast120579

(iii) (119905 119904) Σ rarr 119872 times 119872 is a generalized contact mapcondition of generalized contactmap on119872 is (119889119905)∘120593

1=

1205932∘ (119889119905) this condition on119872 is (119889119904) ∘ 120593

1= minus1205932∘ (119889119904)

Proof (i) hArr (ii) define 120601 = 120579minus119905lowast120579+119904lowast120579 such that 120579 = minus119889120578minus

119868lowast119889120578+120578and(119894119865119889120578) is the twist of (119889120578 119868) and119860 = ker(119889119904)|

119872 We

know fromTheorem 1 that closed multiplicative 2-form 120595 onΣ vanishes if and only if 119868119872 form 119906

120595= 0 that is120595(119883 120572) = 0

such that119883 isin 119879119872 120572 isin 119860 This case can be applied for formswith higher degree that is 3-form 120595 vanishes if and only if120595(119883 119884 120572) = 0

Since 119889120578 and (119889120578)119868

are closed from (35) we get119894119883and119884

(119889(119868lowast119889120578)) = minus119894119873119868(119883119884)

119889120578 Putting 120579 = minus119889120578 minus 119868lowast119889120578 + 120578 and

(119894119865119889120578) we obtain

119894119883and119884

119889 (minus120579 + 120578 and (119894119865119889120578)) = minus119894

119873119868(119883119884)119889120578 (51)

Since 119889120601 = 0 hArr 119889120601(119883 119884 120572) = 0 we have

119889120601 (119883 119884 120572) = 0 lArrrArr 119889120579 (119883 119884 120572) minus 119889 (119905lowast120579) (119883 119884 120572)

+ 119889 (119904lowast120579) (119883 119884 120572) = 0

(52)

On the other hand we obtain

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119889119905 (119883) 119889119905 (119884) 119889119905 (120572)) (53)

If we take 119889119905 = 120588 in (53) for 119860 we get

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119889119905 (119883) 119889119905 (119884) 120588 (120572)) (54)

On the other hand from [17] we know that

119868119889Σ= 119898 ∘ (119905 119868119889

Σ) (55)

Differentiating (55) we obtain

119883 = 119889119905 (119883) (56)

Using (56) in (54) we get

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119883 119884 120588 (120572)) (57)

In a similar way we see that

119889 (119904lowast120579) (119883 119884 120572) = 119889120579 (119889119904 (119883) 119889119904 (119884) 119889119904 (120572)) (58)

Since 120572 isin ker 119889119904 then 119889119904(120572) = 0 Hence 119889(119904lowast120579) = 0 Thuswe obtain

119889120579 (119883 119884 120572) = 119889120579 (119883 119884 120588 (120572)) (59)

Using (51) in (59) we derive

119889 (120578 and (119894119865119889120578)) (119883 119884 120572) + 119889120578 (119873

119868(119883 119884) 120572)

= 119889120579 (119883 119884 120588 (120572)) (60)

Abstract and Applied Analysis 7

On the other hand it is clear that 120601 = 0 hArr 120579 minus 119905lowast120579 + 119904lowast120579 = 0Thus we obtain

120579 (119883 120572) = 120579 (119883 120588 (120572)) (61)

Since 120579 = minus119889120578 minus 119868lowast119889120578 + 120578 and (119894119865119889120578) we get

minus 119889120578 (119883 120572) minus 119889120578 (119868119883 119868120572) + (120578 and (119894119865119889120578)) (119883 120572)

= 120579 (119883 120588 (120572)) (62)

Since Jacobi structure (119865 120587) is integrable it defines a Liealgebroid whose anchor map is 120588 = (120587 119865)

♯ Let us use (120587 119865)♯instead of 120588 in (60) and (62) then we get

119889 (120578 and (119894119865119889120578)) (119883 119884 120572) + 119889120578 (119873

119868(119883 119884) 120572)

= 119889120579 (119883 119884 120587♯(120572) + 119891119865)

(63)

minus 119889120578 (119883 120572) minus 119889120578 (119868119883 119868120572) + (120578 and (119894119865119889120578)) (119883 120572)

= 120579 (119883 120587♯(120572) + 119891119865)

(64)

Since 119889120578(120572119883) = 120572(119883) (119889120578)119868(120572 119883) = 120572(120593119883) from (63) we

have

minus 120572 (119889120578 (119883 119884) 119865) minus 120572 (119873120593(119883 119884))

= 119889120579 (119883 119884 120587♯(120572) + 119891119865)

= 119894119883and119884

119889120579 (120587♯(120572) + 119891119865)

= 119894119883and119884

119889120579 (120587♯(120572)) + 119894

119883and119884119889120579 (119891119865)

= 120587 (120572 119894119883and119884

119889120579)

= minus120572 (120587♯(119894119883and119884

119889120579))

(65)

that is 120572(119889120578(119883 119884)119865 + 119873120593(119883 119884)) = 120572(120587♯(119894

119883and119884119889120579))

Since the above equation holds for all nondegenerate 120572we get

119889120578 (119883 119884) 119865 + 119873120593 (119883 119884) = 120587

♯(119894119883and119884

119889120579) (66)

Then we arrive at

119889120578 (120593119883 120593119884) 119865 + 119873120593 (119883 119884) = 120587

♯(119894119883and119884

119889120579) (67)

On the other hand from (64) we obtain

120572 (119883) + 120572 (1205932119883) minus 120578 (119883) 120572 (119865) = 120587 (120572 119894

119883120579) + 119894119883120579 (119891119865)

= minus120572 (120587♯120579♯119883)

(68)

Thus we get

1205932+ 120587♯120579♯= minus119868119889 + 119865 ⊙ 120578 (69)

Then (i)hArr(ii) follows from (67) and (69)

(ii) hArr (iii) 119889120578+119868lowast119889120578minus120578and119894

119865119889120578 = 119904lowast120579minus119905lowast120579 says that (119905 119904)

is compatible with 2-form 119889120578 Also it is clear that (119905 119904) andbivectors are compatible because Σ is a contact groupoid Wewill check the compatibility of (119905 119904) and (1 1)-tensors Fromcompatibility condition of 119905 and 119904 we get 119889119905 ∘ 119868 = 120593 ∘ 119889119905 and119889119904 ∘ 119868 = minus120593 ∘ 119889119904

For all 120572 isin 119860 119881 isin 120594(Σ) and 119891 isin R we have

119889120578 (120572 119881) = 119889120578 (120572 119889119905119881) (70)

which is equivalent to

120572 (119881) = ⟨119906(119889120578)

(120572) 119889119905119881⟩ (71)

Since 119906(119889120578)

= 119868119889 and 119906((119889120578)119868)

= 120593lowast ∘ 1199011199031such that 119906

((119889120578)119868)(120572) =

120593lowast ∘ 1199011199031(120572 119891) = 120593lowast(120572) we get

⟨120572 120593 (119889119905 (119881))⟩ = 120572 (120593 (119889119905 (119881)))

= 120593lowast120572 (119889119905 (119881))

= ⟨119906((119889120578)119868)

120572 119881⟩

= 119889120578 (120572 119868119881)

= ⟨120572 119889119905 (119868119881)⟩

(72)

Since this equation holds for all 120572 isin 119860 119886(119889119905) = 119889119905(119868) Using119904 = 119905 ∘ 119894

119886 (119889119904 (119881)) = 119886119889 (119905 ∘ 119894) 119881

= 120593 (119889119905 (119889119894 (119881)))

= minus120593 (119889119905 (119881))

= minus119889119904 (119868119881)

(73)

which shows that 119886(119889119904) = minus119889119904(119868) Thus proof is completed

Conflict of Interests

The author declares that she has no conflict of interests

References

[1] C Ehresmann ldquoCategories topologiques et categoriesdifferentiablesrdquo in Colloque de Geometrie Differentielle GlobaleBruxelles Belgium pp 137ndash150 Librairie Universitaire 1959

[2] M Crainic and R L Fernandes ldquoIntegrability of Lie bracketsrdquoAnnals of Mathematics Second Series vol 157 no 2 pp 575ndash620 2003

[3] M Degeratu ldquoProjective limits of Lie groupoidsrdquo AnaleleUniversitatii din Oradea Fascicola Matematica vol 16 pp 183ndash188 2009

[4] M R Farhangdoost and T Nasirzade ldquoGeometrical categoriesof generalized Lie groups and Lie group-groupoidsrdquo IranianJournal of Science and Technology A Science vol 37 no 1 pp69ndash73 2013

[5] C-M Marle ldquoCalculus on Lie algebroids Lie groupoids andPoisson manifoldsrdquo Dissertationes Mathematicae (RozprawyMatematyczne) vol 457 p 57 2008

8 Abstract and Applied Analysis

[6] D S Zhong Z Chen and Z J Liu ldquoOn the existence of globalbisections of Lie groupoidsrdquo Acta Mathematica Sinica (EnglishSeries) vol 25 no 6 pp 1001ndash1014 2009

[7] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiablesCalcul differenetiel dans la categorie des groupoıdesinfinitesimauxrdquo Comptes Rendus de lrsquoAcademie des Sciences Avol 264 pp 245ndash248 1967

[8] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquoThe QuarterlyJournal of Mathematics vol 54 no 3 pp 281ndash308 2003

[9] M Gualtieri Generalized complex geometry [PhD thesis]University of Oxford 2003

[10] D Iglesias-Ponte and A Wade ldquoContact manifolds and gener-alized complex structuresrdquo Journal of Geometry and Physics vol53 no 3 pp 249ndash258 2005

[11] Y S Poon and A Wade ldquoGeneralized contact structuresrdquoJournal of the London Mathematical Society Second Series vol83 no 2 pp 333ndash352 2011

[12] A Wade ldquoLocal structure of generalized contact manifoldsrdquoDifferential Geometry and its Applications vol 30 no 1 pp 124ndash135 2012

[13] M Crainic ldquoGeneralized complex structures and Lie bracketsrdquoBulletin of the Brazilian Mathematical Society New SeriesBoletim da Sociedade Brasileira de Matematica vol 42 no 4pp 559ndash578 2011

[14] M Crainic and C Zhu ldquoIntegrability of Jacobi structuresrdquoAnnales de lrsquoinstitut Fourier vol 57 no 4 pp 1181ndash1216 2007

[15] A A Kirillov ldquoLocal Lie algebrasrdquo Russian MathematicalSurveys vol 31 pp 55ndash75 1976

[16] M Crainic and R L Fernandes ldquoLectures on integrability of Liebracketsrdquo in Lectures on Poisson Geometry vol 17 of Geometryand Topology Monographs pp 1ndash107 Geometry amp TopologyPublications Coventry UK 2011

[17] H BursztynM Crainic AWeinstein and C Zhu ldquoIntegrationof twisted Dirac bracketsrdquo Duke Mathematical Journal vol 123no 3 pp 549ndash607 2004

[18] J Janyska and M Modugno ldquoGeneralized geometrical struc-tures of odd dimensional manifoldsrdquo Journal de MathematiquesPures et Appliquees Neuvieme Serie vol 91 no 2 pp 211ndash2322009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

m(g h)

g h

G

G0

t(m(g h)) = t(g) s(g) = t(h) s(h) = s(m(g h))

Figure 1 Two arrows 119892 and ℎ isin 119866 with the target of ℎ 119905(ℎ) isin 1198660

equal to the source of 119892 119904(119892) isin 1198660 and the composed arrow119898(119892 ℎ)

[5]

groupoid The paper is organized as follows In Section 2 wegathermain definitions and results used in the other sectionsIn Section 3 we first state necessary and sufficient conditionsfor generalized almost contact structure to be integrable andthen we obtain a relation between integrable Jacobi pairs andcontact groupoids defined on generalized manifolds More-over we observe that there is a close relationship between(1 1)-tensors satisfying certain conditions in terms of tensorfields defined on generalized manifolds and multiplicativeforms Finally we find one to one correspondence amonggeneralized contact map source and target maps and theconditions of a generalized contact structure to be integrable

2 Preliminaries

In this section we give basic facts of Jacobi geometry Liegroupoids and Lie algebroids We first recall notions ofcontact manifold and contact groupoid from [5] A contactmanifold is a smooth (odd-dimensional) manifold119872 with 1-form 120578 isin Ω

1(119872) such that 120578and(119889120578)119899

= 0 120578 is called the contactform of119872 Let119866 be a Lie groupoid on119872 and 120578 a form on Liegroupoid 119866 then 120578 is called 119903-multiplicative if

119898lowast120578 = 119901119903

lowast

2(119890minus119903) 119901119903lowast

1120578 + 119901119903

lowast

2120578 (2)

where119901119903119894 119866times119866 rarr 119866 119894 = 1 2 are the canonical projections

and 119903 119866 rarr R 119903(119892ℎ) = 119903(119892) + 119903(ℎ) is a function [14]A contact groupoid over a manifold 119872 is a Lie groupoid 119866

over 119872 together with a contact form 120578 on 119866 such that 120578 is 119903-multiplicative We recall that multiplicative of a 2-form 120596 isdefined by

119898lowast120596 = 119901119903

lowast

1120596 + 119901119903

lowast

2120596 (3)

We now recall the notion of Jacobi manifolds A Jacobimanifold is a smooth manifold 119872 equipped with a bivectorfield 120587 and a vector field 119864 such that

[120587 120587] = minus2119864 and 120587 [119864 120587] = 0 (4)

where [ ] denotes the Schouten bracket In this case (120587 119864)defines a bracket on119862infin(119872R) which is called Jacobi bracketand is given for all 119891 119892 isin 119862infin(119872R) by

119891 119892 = 120587 (119889119891 119889119892) + 119891119864 (119892) minus 119892119864 (119891) (5)

The Jacobi bracket endows119862infin(119872R)with a local Lie algebrastructure in the sense of Kirillov [15]

We now give a relation between Lie algebroid and Liegroupoid more details can be found in [16] Given a Liegroupoid 119866 on 119872 the associated Lie algebroid 119860 = Lie(119866)

has fibres 119860119909= Ker(119889119904)

119909= 119879119909(119866(minus 119909)) for any 119909 isin 119872 Any

120572 isin Γ(119860) extends to a unique right-invariant vector field on119866 which will be denoted by the same letter 120572 The usual Liebracket on vector fields induces the bracket on Γ(119860) and theanchor is given by 120588 = 119889119905 119860 rarr 119879119872

Given a Lie algebroid 119860 an integration of 119860 is a Liegroupoid 119866 together with an isomorphism 119860 cong Lie(119866) Ifsuch a 119866 exists then it is said that 119860 is integrable In contrastwith the case of Lie algebras not every Lie algebroid admitsan integration However if a Lie algebroid is integrable thenthere exists a canonical source-simply connected integration119866 and any other source-simply connected integration issmoothly isomorphic to 119866 From now on we assume that allLie groupoids are source-simply connected

We now recall the notion of 119868119872 form (infinitesimalmultiplicative form) on a Lie algebroid [17] which will beuseful whenwedealwith relations between Lie groupoids andLie algebroids An 119868119872 form on a Lie algebroid 119860 is a bundlemap

119906 119860 997888rarr 119879lowast119872 (6)

satisfying the following properties

(i) ⟨119906(120572) 120588(120573)⟩ = minus⟨119906(120573) 120588(120572)⟩

(ii) 119906([120572 120573]) = L120572(119906(120573)) minusL

120573(119906(120572)) + 119889⟨119906(120572) 120588(120573)⟩

for 120572 120573 isin Γ(119860) where 120588 = 120588119860and ⟨ ⟩ denotes the usual

pairing between a vector space and its dualIf 119860 is a Lie algebroid of a Lie groupoid 119866 then a closed

multiplicative 2-form 120596 on 119866 induces an 119868119872 form 119906120596of 119860 by

⟨119906120596(120572) 119883⟩ = 120596 (120572119883) (7)

For the relationship between 119868119872 form and closed 2-form wehave the following

Theorem 1 (see [17]) If 119860 is an integrable Lie algebroid and if119866 is its integration then120596 997891rarr 119906

120596is a one to one correspondence

between closed multiplicative 2-forms on119866 and IM forms of119860

Finally in this section we give brief information on thenotion of generalized geometry details can be found in [9] Acentral idea in generalized geometry is that119879119872oplus119879

lowast119872 shouldbe thought of as a generalized tangent bundle to manifold119872If 119883 and 120585 denote a vector field and a dual vector field on119872 respectively then we write (119883 120585) (or 119883 + 120585) as a typicalelement of 119879119872 oplus 119879

lowast119872 The Courant bracket of two sections(119883 120585) (119884 120578) of 119879119872 oplus 119879lowast119872 = TM is defined by

⟦(119883 120585) (119884 120578)⟧ = [119883 119884] +L119883120578 minusL

119884120585 minus

1

2119889 (119894119883120578 minus 119894119884120585)

(8)

where119889L119883 and 119894

119883denote exterior derivative Lie derivative

and interior derivative with respect to 119883 respectively TheCourant bracket is antisymmetric but it does not satisfy theJacobi identity Here we use the notations 120573(120587♯120572) = 120587(120572 120573)

Abstract and Applied Analysis 3

and 120596♯(119883)(119884) = 120596(119883 119884) which are defined as 120587♯ 119879lowast119872 rarr

119879119872120596♯ 119879119872 rarr 119879lowast119872 for any 1-forms120572 and120573 2-form120596 and

bivector field 120587 and vector fields119883 and 119884 Also we denote by[ ]120587the bracket on the space of 1-forms on119872 defined by

[120572 120573]120587= L120587♯120572120573 minusL

120587♯120573120572 minus 119889120587 (120572 120573) (9)

3 Lie Groupoids and GeneralizedContact Structures

In this section we first give a characterization for generalizedcontact structures to be integrable then we obtain certainrelationships between generalized contact manifolds andcontact groupoids We recall a generalized almost contactpair and then a generalized almost contact structure

Definition 2 (see [12]) A generalized almost contact pair(I 119865+120578) on a smooth odd-dimensionalmanifold119872 consistsof a bundle endomorphism I of 119879119872 oplus 119879

lowast119872 and a section119865 + 120578 ofTM such that

I +Ilowast= 0 120578 (119865) = 1

I (119865) = 0 I (120578) = 0 I2= minus119868119889 + 119865 ⊙ 120578

(10)

where119865⊙120578(119883+120572) = 120578(119883)119865+120572(119865)120578 for any119883+120572 isin Γ(TM)SinceI has a matrix form

I = [120593 120587♯

120579♯ minus120593lowast] (11)

where 120593 is a (1 1)-tensor 120587 is a bivector field 120579 is a 2-formand 120593lowast 119879lowast119872 rarr 119879lowast119872 is dual of 120593 one sees that ageneralized almost contact pair is equivalent to a quintuplet(119865 120578 120587 120579 120593) where 119865 is a vector field 120578 a 1-form

Definition 3 (see [12]) A generalized almost contact structureon119872 is an equivalent class of such pairs (I 119865 + 120578)

We now present two examples of generalized almostcontact manifolds

Example 4 (see [11]) An (2119899 + 1)-dimensional smoothmanifold 119872 has an almost contact structure (120593 119865 120578) if itadmits a tensor field 120593 of type (1 1) a vector field 119865 and a1-form 120578 satisfying the following compatibility conditions

120593 (119865) = 0 120578 ∘ 120593 = 0

120578 (119865) = 1 1205932= minus119894119889 + 120578 otimes 119865

(12)

Associated with any almost contact structure we have analmost generalized contact structure by setting

I = [120593 0

0 minus120593lowast] (13)

Example 5 (see [11]) On the three-dimensional Heisenberggroup 119867

3 we choose a basis 119883

1 1198832 1198833 and let 120572

1 1205722 1205723

be a dual frame For 119905 = 119903119888+ 119894119903119904 where 119888 = cos V and 119904 = sin Vfor some real number V we define

120593119905=

2119903119888

1 minus 1199032(1198832otimes 1205722+ 1198833otimes 1205723)

120590119905=

1199032 minus 2119903119904 + 1

1 minus 11990321205722and 1205723

120587119905=

1199032 + 2119903119904 + 1

1 minus 11990321198832and 1198833

120578 = 1205721 119865 = 119883

1minus 1198871198832+ 1198861198833

(14)

for any real numbers 119886 119887 We also define

I = [120593119905

120587♯

119905

120590119905

minus120593lowast119905

] (15)

Then J119905= (119865 120578 120601

119905 120587119905120590119905) is a family of generalized almost

contact structures

Given a generalized almost contact pair (I 119865 + 120578) wedefine

119864(10)

= 119890 minus 119894I (119890) | 119890 isin ker 120578 oplus ker119865

119864(01)

= 119890 + 119894I (119890) | 119890 isin ker 120578 oplus ker119865 (16)

The endomorphism I is linearly extended to the complexi-fied bundleTM otimesC It has three eigenvalues namely 120582 = 0120582 = 119894 = radicminus1 and 120582 = minus119894 The corresponding eigenbundlesare 119871119865oplus119871120578119864(10) and119864(01) where 119871

119865and 119871

120578are the complex

vector bundles of rank 1 generated with 119865 and 120578 respectivelyDefine

119871 = 119871119865oplus 119864(10)

119871 = 119871119865oplus 119864(01)

119871lowast= 119871120578oplus 119864(01)

119871lowast

= 119871120578oplus 119864(10)

(17)

Definition 6 (see [11]) Consider a generalized almost contactpair and let 119871 be its associated maximal isotropic subbundleOne says that the generalized almost contact pair is integrableif the space Γ(119871) of sections of 119871 is closed under the Courantbracket that is ⟦Γ(119871) Γ(119871)⟧ sub Γ(119871) In this case thegeneralized almost contact pair is simply called a generalizedcontact pair A generalized contact structure is an equivalenceclass of generalized contact pairs

In the sequel we give necessary and sufficient conditionsfor a generalized almost contact structure to be integrable interms of the above tensor fields We note that the followingresult was stated in [12]

Theorem 7 A generalized almost contact pair correspondingto the quintuplet (119865 120578 120587 120579 120593) is integrable if and only if thefollowing relations are satisfied

(C1)

(a) 1

2[120587 120587] = 119865 and (120587

♯otimes 120587♯) 119889120578

(b) [119865 120587] = minus119865 and 120587♯L119865120578

(18)

4 Abstract and Applied Analysis

(C2)

120593120587♯= 120587♯120593lowast

120593lowast[120572 120573]

120587= L120587♯120572120593lowast120573 minusL

120587♯120573120593lowast120572 minus 119889120587 (120593

lowast120572 120573)

(19)

(C3)

1205932+ 120587♯120579♯= minus119868119889 + 119865 ⊙ 120578

119873120593 (119883 119884) + 119889120578 (120593119883 120593119884) 119865 = 120587

♯(119894119883and119884

119889120579)

(20)

(C4)

120593lowast120579♯= 120579♯120593

119889120579120593 (119883 119884 119885)

= 119889120579 (120593119883 119884 119885) + 119889120579 (119883 120593119884 119885) + 119889120579 (119883 119884 120593119885)

(21)

(C5)

L119865120593 = 0 L

119865120579 = 0 (22)

where

[120572 120573]120587= L120587♯120572120573 minusL

120587♯120573120572 minus 119889120587 (120572 120573)

120579120593 (119883 119884) = 120579 (120593119883 119884)

(23)

We note that if (11) is a generalized contact structure then

I = [120593 minus120587♯

minus120579♯

minus120593lowast] (24)

is also a generalized contact structureI is called the oppositeofI In this paper we denote a generalized contact manifoldendowed withI by119872

As an analogue of aHitchin pair on a generalized complexmanifold a Hitchin pair on a generalized almost contactmanifold 119872 is a pair (119889120578 120593) consisting of a contact form120578 and a (1 1)-tensor 120593 with the property that 119889120578 and 120593

commute (ie 119889120578(119883 120593119884) = 119889120578(120593119883 119884)) We note that sincea generalized almost contact structure is equivalent to ageneralized almost complex structure on119872timesR the bivectorfield 120587 of the generalized almost contact structure is notnondegenerate in general But we emphasize that we areputting this condition for restricted case

Lemma 8 Let119872 be a generalized almost contact manifold If120587 is a nondegenerate bivector field on 119879119872

lowast-119878119901119886119899120578 119889120578 is theinverse 2-form (defined by (119889120578)

♯= (120587♯)

minus1) and120587 satisfies (20)then 120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894

119865119889120578) if and only if 119889120578(1205932119883119884) =

120593lowast119889120578(119883 119884)

Proof For 119883 isin 120594(119872) we apply (119889120578)♯to (20) and using the

dual structure 120593lowast we have

(119889120578)♯1205932(119883)

= minus(119889120578)♯(119883) minus (119889120578)

♯(120587♯120579♯ (119883)) + (119889120578)

♯(120578 (119883) 119865)

119889120578 (1205932119883119884) = minus119889120578 (119883 119884) minus 120579

♯(119883) (119884) + 119889120578 (120578 (119883) 119865 119884)

(25)

We obtain

120593lowast119889120578 (119883 119884) = minus119889120578 (119883 119884) minus 120579 (119883 119884) + 120578 (119883) 119889120578 (119865 119884)

(26)

Since (26) holds for all119883 and 119884 we get

120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894

119865119889120578) (27)

In a similar way one can get the converse

From now on when we mention a nondegenerate bivec-tor field 120587 we mean it is nondegenerate on 119879119872lowast-Span120578We note that if 119889120578 is the inverse 2-form of 120587 nondegenerate120587 on 119879119872

lowast-Span120578 implies that 119889120578 is also nondegenerate on119879119872-Span119865

We say that 2-form 120579 is the twist of Hitchin pair (119889120578 120593)Note that in this case 120593 is neither an almost contact structurenor torsion(119873

120593) free

Lemma 9 Let (119872 120578 120593 119865) be an almost contact manifold 119889120578and 120593 commute if and only if 119889120578 + 120593lowast119889120578 = 120578 and (119894

119865119889120578)

Proof We will only prove the sufficient condition We have

120593lowast119889120578 (119883 119884) + 119889120578 (119883 119884) minus 120578 and (119894

119865119889120578) (119883 119884) = 0 (28)

Since 120593lowast is dual contact structure we get

119889120578 (120593119883 120593119884) + 119889120578 (119883 119884) minus 120578 (119883) (119894119865119889120578) (119884) = 0 (29)

Substituting 120593119883 by119883 and using contact structure property

119889120578 (minus119883 + 120578 (119883) 119865 120593119884) + 119889120578 (120593119883 119884) = 0 (30)

Hence we obtain

minus119889120578 (119883 120593119884) + 119889120578 (120593119883 119884) = 0 (31)

which shows that 119889120578 and 120593 commute The converse is clear

Next we see that (C1) is satisfied automatically when onechooses 119889120578 as the 2-form which is the inverse of 120587 defined by(119889120578)♯= (120587♯)

minus1

Lemma 10 Let120587 be a nondegenerate bivector on a generalizedalmost contact manifold119872 and 119889120578 the inverse 2-form (definedby (119889120578)

♯= (120587♯)

minus1) Then 120587 satisfies (C1)

Proof Since 119889120578 is a closed form it is obvious due to [13Lemma 27]

Abstract and Applied Analysis 5

Definition 11 (see [14]) The Lie algebroid of the Jacobimanifold (119872 120587 119865) is 119879lowast119872 oplus R with the anchor 120588 119879lowast119872 oplus

R rarr 119879119872 given by

120588 (120596 120582) = (120587 119865)♯(120596 120582) = 120587

♯(120596) + 120582119865 (32)

and the bracket

[(120596 0) (120578 0)] = ([120596 120578]120587 0) minus (119894

119865(120596 and 120578) 120587 (120596 120578))

[(0 1) (120596 0)] = (L(119865)

120596 0) (33)

The associated groupoid

Σ (119872) = 119866 (119879lowast119872 oplusR) (34)

is called contact groupoid of the Jacobi manifold 119872 We saythat 119872 is integrable as a Jacobi manifold if the associatedalgebroid 119879lowast119872 oplus R is integrable (or equivalently if Σ(119872) issmooth)

Thus we have the following result which shows that thereis a close relationship between the condition (C1) and acontact groupoid

Theorem 12 Let 119872 be a generalized almost contact manifoldand 120578 a contact form There is a 1-1 correspondence between

(i) integrable Jacobi pair (119865 120587) on 119872 (ie (119865 120587) issatisfying (C1) integrable)

(ii) contact groupoids (Σ 120578) over119872

Proof Since (120578 119889120578) is a contact pair and (119865 120587) satisfies(C1) then (119865 120587) is an integrable Jacobi pair [18] FromDefinition 11 one sees that a contact groupoid is obtainedfrom an integrable Jacobi pair

The converse is clear

We now give the conditions for (C2) in terms of 119889120578 and120593

Lemma 13 Let 119872 be a generalized almost contact manifoldand 119889120578 a 2-form Given a nondegenerate bivector 120587 on119879119872lowast-120578 (ie 120587♯ = ((119889120578)

♯)minus1) and a map 120593 119879119872 rarr 119879119872

then 120587 and 120593 satisfy (C2) if and only if 119889120578 and 120593 commute

We now give a correspondence between generalizedcontact structures with nondegenerate 120587 and Hitchin pairs(119889120578 120593)

Proposition 14 There is a one to one correspondence betweengeneralized contact structures given by (11) with 120587 nonde-generate and Hitchin pairs (119889120578 120593) such that 119889120578(119883 119884) =

119889120578(120593119883 120593119884) In this correspondence 120587 is the inverse of 119889120578 and120579 is the twist of the Hitchin pair (119889120578 120593)

Proof Since (119889120578 120593) is Hitchin pair 119889120578 and (119889120578)120593are closed

By using the following equation (see [13])

119894119873120593(119883119884)

(119889120578) = 119894120593119883and119884+119883and120593119884

(119889(119889120578)120593) minus 119894120593119883and120593119884

(119889 (119889120578))

minus 119894119883and119884

(119889 (120593lowast119889120578))

(35)

we get

119894119873120593(119883119884)

(119889120578) = minus119894119883and119884

(119889 (120593lowast119889120578)) (36)

Since 120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894119865119889120578) we derive

(119889120578)♯(119873120593(119883 119884)) = minus119894

119883and119884(119889 (minus120579 + 120578 and (119894

119865119889120578))) (37)

Applying 120587♯ to (37) then we get

119873120593(119883 119884) = minus120587

♯(119894119883and119884

(119889 (minus120579 + 120578 and (119894119865119889120578))))

119873120593(119883 119884) = 120587

♯(119894119883and119884

(119889120579) minus 120587♯(119894119883and119884

119889120578 and (119894119865119889120578)))

119873120593(119883 119884) = 120587

♯(119894(119883and119884)

119889120579) minus 119889120578 (119883 119884) 119865

(38)

By assumption we have

119889120578 (119883 119884) = 119889120578 (120593119883 120593119884) (39)

Putting this equation into (38) we obtain

119873120593(119883 119884) = 120587

♯(119894(119883and119884)

119889120579) minus 119889120578 (120593119883 120593119884) 119865 (40)

which is the second equation of (C3) Now we show that120593lowast120579♯= 120579♯120593 From (26) we obtain

120593lowast120579♯= 120593lowast(minus(119889120578)

♯minus (120593lowast(119889120578))♯+ (120578 and (119894

119865(119889120578)))

♯) (41)

Hence we have

120593lowast120579♯= minus(119889120578)

♯120593 minus (120593

lowast(119889120578))♯120593 + (120578 and (119894

119865(119889120578)))

♯120593 (42)

From definition of twist we get

120593lowast120579♯= 120579♯120593 (43)

This equation is the first equation of (C4) Now wewill obtain

119889120579120593(119883 119884 119885)

= 119889120579 (120593119883 119884 119885) + 119889120579 (119883 120593119884 119885) + 119889120579 (119883 119884 120593119885)

(44)

which is second equation of (C4) Writing the equation as

119894119883and119884

(119889120579120593) = 119894120593119883and119884+119883and120593119884 (119889120579) + 120593

lowast(119894119883and119884 (119889120579)) (45)

and since 120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894

119865119889120578) then we should find

119894119883and119884

(119889 (minus(120593lowast(119889120578))120593+ (120578 and (119894

119865(119889120578)))

120593))

= 119894120593119883and119884+119883and120593119884

(119889 (minus120593lowast119889120578 + 120578 and (119894

119865119889120578)))

+ 120593lowast(119894119883and119884

(119889 (minus120593lowast119889120578 + 120578 and (119894

119865119889120578))))

(46)

A straightforward computation shows that

119894119883and119884

(119889 ((120593lowast(119889120578))120593)) = 119894

120593119883and119884+119883and120593119884(119889 (120593lowast119889120578))

+ 120593lowast(119894119883and119884

(119889 (120593lowast119889120578)))

(47)

6 Abstract and Applied Analysis

Using (35) then we get

119894119883and119884

(119889 ((120593lowast(119889120578))120593))

= 119894120593119883and119884+119883and120593119884

(119889 (120593lowast119889120578)) minus 119894

119873120593(119883119884)(119889120578)120593

(48)

Since 119894119873(119883119884)

((119889120578)120593) = 120593lowast119894

119873(119883119884)(119889120578) applying 119894

119883and119884119889(120593lowast119889120578)

= minus119894119873120593(119883119884)

(119889120578) to (48) we have

119894119883and119884

(119889 ((120593lowast(119889120578))120593))

= 119894120593119883and119884+119883and120593119884

(119889 (120593lowast119889120578)) + 120593

lowast(119894119883and119884

(119889 (120593lowast119889120578)))

(49)

In a similar way we can obtain (C5)The converse is clear from Lemmas 10 and 13

We note that similar to 2-forms given a Lie groupoid 119866a (1 1)-tensor 119869 119879119866 rarr 119879119866 is called multiplicative [13] iffor any (119892 ℎ) isin 119866 times 119866 and any V

119892isin 119879119892119866 119908ℎisin 119879ℎ119866 such that

(V119892 119908ℎ) is tangent to 119866 times 119866 at (119892 ℎ) so is (119869V

119892 119869119908ℎ) and

(119889119898)119892ℎ

(119869V119892 119869119908ℎ) = 119869 ((119889119898)

119892ℎ(V119892 119908ℎ)) (50)

Let (119872 120578) be a contact manifold Then it is easy to see thatthere is a one to one correspondence between (1 1)-tensors120593 commuting with 119889120578 and 2-forms on119872 On the other handit is easy to see that (C2) is equivalent to the fact that 120593lowast ∘ 119901119903

1

is an 119868119872 form on the Lie algebroid119879lowast119872oplusR associated Jacobistructure (119865 120587) Thus from the above discussion Lemma 13andTheorem 1 one can concludewith the following theorem

Theorem 15 Let119872 be a generalized almost contact manifoldLet (119865 120587) be an integrable Jacobi structure on 119872 and (Σ 120578)

a contact groupoid over 119872 Then there is a natural 1-1correspondence between

(i) (1 1)-tensors 120593 on119872 satisfying (C2)

(ii) multiplicative (1 1)-tensors 119868 on Σ with the propertythat (119868 119889120578) is a Hitchin pair

We recall the notion of generalized contact map betweengeneralized contact manifolds This notion is similar to thegeneralized holomorphic map given in [13]

Let (119872119894I119894) 119894 = 1 2 be two generalized contact

manifolds and let 120593119894 120587119894 120579119894be the components of I

119894in the

matrix representation (11) A map 119891 1198721

rarr 1198722is called

generalized contact if and only if 119891 maps 1205931into 120593

2 1198651into

1198652 and 120587

1into 120587

2 119891lowast1205792= 1205791and (119889119891) ∘ 120593

1= 1205932∘ (119889119891)

We now state and prove themain result of this paperThisresult gives equivalent assertions between the condition (C3)twist 120579 of (119889120578 119868) and contact maps for a contact groupoidover119872

Theorem 16 Let 119872 be a generalized almost contact manifoldand (Σ 120578 119868) an induced contact groupoid over 119872 with theinduced multiplicative (1 1)-tensor Assume that ((119865 120587) 119868)

satisfy (C1) (C2) with integrable (119865 120587) Then for a 120579 2-formon119872 the following assertions are equivalent

(i) (C3) is satisfied(ii) 119889120578 + 119868

lowast119889120578 minus 120578 and (119894119865119889120578) = 119904lowast120579 minus 119905lowast120579

(iii) (119905 119904) Σ rarr 119872 times 119872 is a generalized contact mapcondition of generalized contactmap on119872 is (119889119905)∘120593

1=

1205932∘ (119889119905) this condition on119872 is (119889119904) ∘ 120593

1= minus1205932∘ (119889119904)

Proof (i) hArr (ii) define 120601 = 120579minus119905lowast120579+119904lowast120579 such that 120579 = minus119889120578minus

119868lowast119889120578+120578and(119894119865119889120578) is the twist of (119889120578 119868) and119860 = ker(119889119904)|

119872 We

know fromTheorem 1 that closed multiplicative 2-form 120595 onΣ vanishes if and only if 119868119872 form 119906

120595= 0 that is120595(119883 120572) = 0

such that119883 isin 119879119872 120572 isin 119860 This case can be applied for formswith higher degree that is 3-form 120595 vanishes if and only if120595(119883 119884 120572) = 0

Since 119889120578 and (119889120578)119868

are closed from (35) we get119894119883and119884

(119889(119868lowast119889120578)) = minus119894119873119868(119883119884)

119889120578 Putting 120579 = minus119889120578 minus 119868lowast119889120578 + 120578 and

(119894119865119889120578) we obtain

119894119883and119884

119889 (minus120579 + 120578 and (119894119865119889120578)) = minus119894

119873119868(119883119884)119889120578 (51)

Since 119889120601 = 0 hArr 119889120601(119883 119884 120572) = 0 we have

119889120601 (119883 119884 120572) = 0 lArrrArr 119889120579 (119883 119884 120572) minus 119889 (119905lowast120579) (119883 119884 120572)

+ 119889 (119904lowast120579) (119883 119884 120572) = 0

(52)

On the other hand we obtain

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119889119905 (119883) 119889119905 (119884) 119889119905 (120572)) (53)

If we take 119889119905 = 120588 in (53) for 119860 we get

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119889119905 (119883) 119889119905 (119884) 120588 (120572)) (54)

On the other hand from [17] we know that

119868119889Σ= 119898 ∘ (119905 119868119889

Σ) (55)

Differentiating (55) we obtain

119883 = 119889119905 (119883) (56)

Using (56) in (54) we get

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119883 119884 120588 (120572)) (57)

In a similar way we see that

119889 (119904lowast120579) (119883 119884 120572) = 119889120579 (119889119904 (119883) 119889119904 (119884) 119889119904 (120572)) (58)

Since 120572 isin ker 119889119904 then 119889119904(120572) = 0 Hence 119889(119904lowast120579) = 0 Thuswe obtain

119889120579 (119883 119884 120572) = 119889120579 (119883 119884 120588 (120572)) (59)

Using (51) in (59) we derive

119889 (120578 and (119894119865119889120578)) (119883 119884 120572) + 119889120578 (119873

119868(119883 119884) 120572)

= 119889120579 (119883 119884 120588 (120572)) (60)

Abstract and Applied Analysis 7

On the other hand it is clear that 120601 = 0 hArr 120579 minus 119905lowast120579 + 119904lowast120579 = 0Thus we obtain

120579 (119883 120572) = 120579 (119883 120588 (120572)) (61)

Since 120579 = minus119889120578 minus 119868lowast119889120578 + 120578 and (119894119865119889120578) we get

minus 119889120578 (119883 120572) minus 119889120578 (119868119883 119868120572) + (120578 and (119894119865119889120578)) (119883 120572)

= 120579 (119883 120588 (120572)) (62)

Since Jacobi structure (119865 120587) is integrable it defines a Liealgebroid whose anchor map is 120588 = (120587 119865)

♯ Let us use (120587 119865)♯instead of 120588 in (60) and (62) then we get

119889 (120578 and (119894119865119889120578)) (119883 119884 120572) + 119889120578 (119873

119868(119883 119884) 120572)

= 119889120579 (119883 119884 120587♯(120572) + 119891119865)

(63)

minus 119889120578 (119883 120572) minus 119889120578 (119868119883 119868120572) + (120578 and (119894119865119889120578)) (119883 120572)

= 120579 (119883 120587♯(120572) + 119891119865)

(64)

Since 119889120578(120572119883) = 120572(119883) (119889120578)119868(120572 119883) = 120572(120593119883) from (63) we

have

minus 120572 (119889120578 (119883 119884) 119865) minus 120572 (119873120593(119883 119884))

= 119889120579 (119883 119884 120587♯(120572) + 119891119865)

= 119894119883and119884

119889120579 (120587♯(120572) + 119891119865)

= 119894119883and119884

119889120579 (120587♯(120572)) + 119894

119883and119884119889120579 (119891119865)

= 120587 (120572 119894119883and119884

119889120579)

= minus120572 (120587♯(119894119883and119884

119889120579))

(65)

that is 120572(119889120578(119883 119884)119865 + 119873120593(119883 119884)) = 120572(120587♯(119894

119883and119884119889120579))

Since the above equation holds for all nondegenerate 120572we get

119889120578 (119883 119884) 119865 + 119873120593 (119883 119884) = 120587

♯(119894119883and119884

119889120579) (66)

Then we arrive at

119889120578 (120593119883 120593119884) 119865 + 119873120593 (119883 119884) = 120587

♯(119894119883and119884

119889120579) (67)

On the other hand from (64) we obtain

120572 (119883) + 120572 (1205932119883) minus 120578 (119883) 120572 (119865) = 120587 (120572 119894

119883120579) + 119894119883120579 (119891119865)

= minus120572 (120587♯120579♯119883)

(68)

Thus we get

1205932+ 120587♯120579♯= minus119868119889 + 119865 ⊙ 120578 (69)

Then (i)hArr(ii) follows from (67) and (69)

(ii) hArr (iii) 119889120578+119868lowast119889120578minus120578and119894

119865119889120578 = 119904lowast120579minus119905lowast120579 says that (119905 119904)

is compatible with 2-form 119889120578 Also it is clear that (119905 119904) andbivectors are compatible because Σ is a contact groupoid Wewill check the compatibility of (119905 119904) and (1 1)-tensors Fromcompatibility condition of 119905 and 119904 we get 119889119905 ∘ 119868 = 120593 ∘ 119889119905 and119889119904 ∘ 119868 = minus120593 ∘ 119889119904

For all 120572 isin 119860 119881 isin 120594(Σ) and 119891 isin R we have

119889120578 (120572 119881) = 119889120578 (120572 119889119905119881) (70)

which is equivalent to

120572 (119881) = ⟨119906(119889120578)

(120572) 119889119905119881⟩ (71)

Since 119906(119889120578)

= 119868119889 and 119906((119889120578)119868)

= 120593lowast ∘ 1199011199031such that 119906

((119889120578)119868)(120572) =

120593lowast ∘ 1199011199031(120572 119891) = 120593lowast(120572) we get

⟨120572 120593 (119889119905 (119881))⟩ = 120572 (120593 (119889119905 (119881)))

= 120593lowast120572 (119889119905 (119881))

= ⟨119906((119889120578)119868)

120572 119881⟩

= 119889120578 (120572 119868119881)

= ⟨120572 119889119905 (119868119881)⟩

(72)

Since this equation holds for all 120572 isin 119860 119886(119889119905) = 119889119905(119868) Using119904 = 119905 ∘ 119894

119886 (119889119904 (119881)) = 119886119889 (119905 ∘ 119894) 119881

= 120593 (119889119905 (119889119894 (119881)))

= minus120593 (119889119905 (119881))

= minus119889119904 (119868119881)

(73)

which shows that 119886(119889119904) = minus119889119904(119868) Thus proof is completed

Conflict of Interests

The author declares that she has no conflict of interests

References

[1] C Ehresmann ldquoCategories topologiques et categoriesdifferentiablesrdquo in Colloque de Geometrie Differentielle GlobaleBruxelles Belgium pp 137ndash150 Librairie Universitaire 1959

[2] M Crainic and R L Fernandes ldquoIntegrability of Lie bracketsrdquoAnnals of Mathematics Second Series vol 157 no 2 pp 575ndash620 2003

[3] M Degeratu ldquoProjective limits of Lie groupoidsrdquo AnaleleUniversitatii din Oradea Fascicola Matematica vol 16 pp 183ndash188 2009

[4] M R Farhangdoost and T Nasirzade ldquoGeometrical categoriesof generalized Lie groups and Lie group-groupoidsrdquo IranianJournal of Science and Technology A Science vol 37 no 1 pp69ndash73 2013

[5] C-M Marle ldquoCalculus on Lie algebroids Lie groupoids andPoisson manifoldsrdquo Dissertationes Mathematicae (RozprawyMatematyczne) vol 457 p 57 2008

8 Abstract and Applied Analysis

[6] D S Zhong Z Chen and Z J Liu ldquoOn the existence of globalbisections of Lie groupoidsrdquo Acta Mathematica Sinica (EnglishSeries) vol 25 no 6 pp 1001ndash1014 2009

[7] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiablesCalcul differenetiel dans la categorie des groupoıdesinfinitesimauxrdquo Comptes Rendus de lrsquoAcademie des Sciences Avol 264 pp 245ndash248 1967

[8] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquoThe QuarterlyJournal of Mathematics vol 54 no 3 pp 281ndash308 2003

[9] M Gualtieri Generalized complex geometry [PhD thesis]University of Oxford 2003

[10] D Iglesias-Ponte and A Wade ldquoContact manifolds and gener-alized complex structuresrdquo Journal of Geometry and Physics vol53 no 3 pp 249ndash258 2005

[11] Y S Poon and A Wade ldquoGeneralized contact structuresrdquoJournal of the London Mathematical Society Second Series vol83 no 2 pp 333ndash352 2011

[12] A Wade ldquoLocal structure of generalized contact manifoldsrdquoDifferential Geometry and its Applications vol 30 no 1 pp 124ndash135 2012

[13] M Crainic ldquoGeneralized complex structures and Lie bracketsrdquoBulletin of the Brazilian Mathematical Society New SeriesBoletim da Sociedade Brasileira de Matematica vol 42 no 4pp 559ndash578 2011

[14] M Crainic and C Zhu ldquoIntegrability of Jacobi structuresrdquoAnnales de lrsquoinstitut Fourier vol 57 no 4 pp 1181ndash1216 2007

[15] A A Kirillov ldquoLocal Lie algebrasrdquo Russian MathematicalSurveys vol 31 pp 55ndash75 1976

[16] M Crainic and R L Fernandes ldquoLectures on integrability of Liebracketsrdquo in Lectures on Poisson Geometry vol 17 of Geometryand Topology Monographs pp 1ndash107 Geometry amp TopologyPublications Coventry UK 2011

[17] H BursztynM Crainic AWeinstein and C Zhu ldquoIntegrationof twisted Dirac bracketsrdquo Duke Mathematical Journal vol 123no 3 pp 549ndash607 2004

[18] J Janyska and M Modugno ldquoGeneralized geometrical struc-tures of odd dimensional manifoldsrdquo Journal de MathematiquesPures et Appliquees Neuvieme Serie vol 91 no 2 pp 211ndash2322009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

and 120596♯(119883)(119884) = 120596(119883 119884) which are defined as 120587♯ 119879lowast119872 rarr

119879119872120596♯ 119879119872 rarr 119879lowast119872 for any 1-forms120572 and120573 2-form120596 and

bivector field 120587 and vector fields119883 and 119884 Also we denote by[ ]120587the bracket on the space of 1-forms on119872 defined by

[120572 120573]120587= L120587♯120572120573 minusL

120587♯120573120572 minus 119889120587 (120572 120573) (9)

3 Lie Groupoids and GeneralizedContact Structures

In this section we first give a characterization for generalizedcontact structures to be integrable then we obtain certainrelationships between generalized contact manifolds andcontact groupoids We recall a generalized almost contactpair and then a generalized almost contact structure

Definition 2 (see [12]) A generalized almost contact pair(I 119865+120578) on a smooth odd-dimensionalmanifold119872 consistsof a bundle endomorphism I of 119879119872 oplus 119879

lowast119872 and a section119865 + 120578 ofTM such that

I +Ilowast= 0 120578 (119865) = 1

I (119865) = 0 I (120578) = 0 I2= minus119868119889 + 119865 ⊙ 120578

(10)

where119865⊙120578(119883+120572) = 120578(119883)119865+120572(119865)120578 for any119883+120572 isin Γ(TM)SinceI has a matrix form

I = [120593 120587♯

120579♯ minus120593lowast] (11)

where 120593 is a (1 1)-tensor 120587 is a bivector field 120579 is a 2-formand 120593lowast 119879lowast119872 rarr 119879lowast119872 is dual of 120593 one sees that ageneralized almost contact pair is equivalent to a quintuplet(119865 120578 120587 120579 120593) where 119865 is a vector field 120578 a 1-form

Definition 3 (see [12]) A generalized almost contact structureon119872 is an equivalent class of such pairs (I 119865 + 120578)

We now present two examples of generalized almostcontact manifolds

Example 4 (see [11]) An (2119899 + 1)-dimensional smoothmanifold 119872 has an almost contact structure (120593 119865 120578) if itadmits a tensor field 120593 of type (1 1) a vector field 119865 and a1-form 120578 satisfying the following compatibility conditions

120593 (119865) = 0 120578 ∘ 120593 = 0

120578 (119865) = 1 1205932= minus119894119889 + 120578 otimes 119865

(12)

Associated with any almost contact structure we have analmost generalized contact structure by setting

I = [120593 0

0 minus120593lowast] (13)

Example 5 (see [11]) On the three-dimensional Heisenberggroup 119867

3 we choose a basis 119883

1 1198832 1198833 and let 120572

1 1205722 1205723

be a dual frame For 119905 = 119903119888+ 119894119903119904 where 119888 = cos V and 119904 = sin Vfor some real number V we define

120593119905=

2119903119888

1 minus 1199032(1198832otimes 1205722+ 1198833otimes 1205723)

120590119905=

1199032 minus 2119903119904 + 1

1 minus 11990321205722and 1205723

120587119905=

1199032 + 2119903119904 + 1

1 minus 11990321198832and 1198833

120578 = 1205721 119865 = 119883

1minus 1198871198832+ 1198861198833

(14)

for any real numbers 119886 119887 We also define

I = [120593119905

120587♯

119905

120590119905

minus120593lowast119905

] (15)

Then J119905= (119865 120578 120601

119905 120587119905120590119905) is a family of generalized almost

contact structures

Given a generalized almost contact pair (I 119865 + 120578) wedefine

119864(10)

= 119890 minus 119894I (119890) | 119890 isin ker 120578 oplus ker119865

119864(01)

= 119890 + 119894I (119890) | 119890 isin ker 120578 oplus ker119865 (16)

The endomorphism I is linearly extended to the complexi-fied bundleTM otimesC It has three eigenvalues namely 120582 = 0120582 = 119894 = radicminus1 and 120582 = minus119894 The corresponding eigenbundlesare 119871119865oplus119871120578119864(10) and119864(01) where 119871

119865and 119871

120578are the complex

vector bundles of rank 1 generated with 119865 and 120578 respectivelyDefine

119871 = 119871119865oplus 119864(10)

119871 = 119871119865oplus 119864(01)

119871lowast= 119871120578oplus 119864(01)

119871lowast

= 119871120578oplus 119864(10)

(17)

Definition 6 (see [11]) Consider a generalized almost contactpair and let 119871 be its associated maximal isotropic subbundleOne says that the generalized almost contact pair is integrableif the space Γ(119871) of sections of 119871 is closed under the Courantbracket that is ⟦Γ(119871) Γ(119871)⟧ sub Γ(119871) In this case thegeneralized almost contact pair is simply called a generalizedcontact pair A generalized contact structure is an equivalenceclass of generalized contact pairs

In the sequel we give necessary and sufficient conditionsfor a generalized almost contact structure to be integrable interms of the above tensor fields We note that the followingresult was stated in [12]

Theorem 7 A generalized almost contact pair correspondingto the quintuplet (119865 120578 120587 120579 120593) is integrable if and only if thefollowing relations are satisfied

(C1)

(a) 1

2[120587 120587] = 119865 and (120587

♯otimes 120587♯) 119889120578

(b) [119865 120587] = minus119865 and 120587♯L119865120578

(18)

4 Abstract and Applied Analysis

(C2)

120593120587♯= 120587♯120593lowast

120593lowast[120572 120573]

120587= L120587♯120572120593lowast120573 minusL

120587♯120573120593lowast120572 minus 119889120587 (120593

lowast120572 120573)

(19)

(C3)

1205932+ 120587♯120579♯= minus119868119889 + 119865 ⊙ 120578

119873120593 (119883 119884) + 119889120578 (120593119883 120593119884) 119865 = 120587

♯(119894119883and119884

119889120579)

(20)

(C4)

120593lowast120579♯= 120579♯120593

119889120579120593 (119883 119884 119885)

= 119889120579 (120593119883 119884 119885) + 119889120579 (119883 120593119884 119885) + 119889120579 (119883 119884 120593119885)

(21)

(C5)

L119865120593 = 0 L

119865120579 = 0 (22)

where

[120572 120573]120587= L120587♯120572120573 minusL

120587♯120573120572 minus 119889120587 (120572 120573)

120579120593 (119883 119884) = 120579 (120593119883 119884)

(23)

We note that if (11) is a generalized contact structure then

I = [120593 minus120587♯

minus120579♯

minus120593lowast] (24)

is also a generalized contact structureI is called the oppositeofI In this paper we denote a generalized contact manifoldendowed withI by119872

As an analogue of aHitchin pair on a generalized complexmanifold a Hitchin pair on a generalized almost contactmanifold 119872 is a pair (119889120578 120593) consisting of a contact form120578 and a (1 1)-tensor 120593 with the property that 119889120578 and 120593

commute (ie 119889120578(119883 120593119884) = 119889120578(120593119883 119884)) We note that sincea generalized almost contact structure is equivalent to ageneralized almost complex structure on119872timesR the bivectorfield 120587 of the generalized almost contact structure is notnondegenerate in general But we emphasize that we areputting this condition for restricted case

Lemma 8 Let119872 be a generalized almost contact manifold If120587 is a nondegenerate bivector field on 119879119872

lowast-119878119901119886119899120578 119889120578 is theinverse 2-form (defined by (119889120578)

♯= (120587♯)

minus1) and120587 satisfies (20)then 120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894

119865119889120578) if and only if 119889120578(1205932119883119884) =

120593lowast119889120578(119883 119884)

Proof For 119883 isin 120594(119872) we apply (119889120578)♯to (20) and using the

dual structure 120593lowast we have

(119889120578)♯1205932(119883)

= minus(119889120578)♯(119883) minus (119889120578)

♯(120587♯120579♯ (119883)) + (119889120578)

♯(120578 (119883) 119865)

119889120578 (1205932119883119884) = minus119889120578 (119883 119884) minus 120579

♯(119883) (119884) + 119889120578 (120578 (119883) 119865 119884)

(25)

We obtain

120593lowast119889120578 (119883 119884) = minus119889120578 (119883 119884) minus 120579 (119883 119884) + 120578 (119883) 119889120578 (119865 119884)

(26)

Since (26) holds for all119883 and 119884 we get

120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894

119865119889120578) (27)

In a similar way one can get the converse

From now on when we mention a nondegenerate bivec-tor field 120587 we mean it is nondegenerate on 119879119872lowast-Span120578We note that if 119889120578 is the inverse 2-form of 120587 nondegenerate120587 on 119879119872

lowast-Span120578 implies that 119889120578 is also nondegenerate on119879119872-Span119865

We say that 2-form 120579 is the twist of Hitchin pair (119889120578 120593)Note that in this case 120593 is neither an almost contact structurenor torsion(119873

120593) free

Lemma 9 Let (119872 120578 120593 119865) be an almost contact manifold 119889120578and 120593 commute if and only if 119889120578 + 120593lowast119889120578 = 120578 and (119894

119865119889120578)

Proof We will only prove the sufficient condition We have

120593lowast119889120578 (119883 119884) + 119889120578 (119883 119884) minus 120578 and (119894

119865119889120578) (119883 119884) = 0 (28)

Since 120593lowast is dual contact structure we get

119889120578 (120593119883 120593119884) + 119889120578 (119883 119884) minus 120578 (119883) (119894119865119889120578) (119884) = 0 (29)

Substituting 120593119883 by119883 and using contact structure property

119889120578 (minus119883 + 120578 (119883) 119865 120593119884) + 119889120578 (120593119883 119884) = 0 (30)

Hence we obtain

minus119889120578 (119883 120593119884) + 119889120578 (120593119883 119884) = 0 (31)

which shows that 119889120578 and 120593 commute The converse is clear

Next we see that (C1) is satisfied automatically when onechooses 119889120578 as the 2-form which is the inverse of 120587 defined by(119889120578)♯= (120587♯)

minus1

Lemma 10 Let120587 be a nondegenerate bivector on a generalizedalmost contact manifold119872 and 119889120578 the inverse 2-form (definedby (119889120578)

♯= (120587♯)

minus1) Then 120587 satisfies (C1)

Proof Since 119889120578 is a closed form it is obvious due to [13Lemma 27]

Abstract and Applied Analysis 5

Definition 11 (see [14]) The Lie algebroid of the Jacobimanifold (119872 120587 119865) is 119879lowast119872 oplus R with the anchor 120588 119879lowast119872 oplus

R rarr 119879119872 given by

120588 (120596 120582) = (120587 119865)♯(120596 120582) = 120587

♯(120596) + 120582119865 (32)

and the bracket

[(120596 0) (120578 0)] = ([120596 120578]120587 0) minus (119894

119865(120596 and 120578) 120587 (120596 120578))

[(0 1) (120596 0)] = (L(119865)

120596 0) (33)

The associated groupoid

Σ (119872) = 119866 (119879lowast119872 oplusR) (34)

is called contact groupoid of the Jacobi manifold 119872 We saythat 119872 is integrable as a Jacobi manifold if the associatedalgebroid 119879lowast119872 oplus R is integrable (or equivalently if Σ(119872) issmooth)

Thus we have the following result which shows that thereis a close relationship between the condition (C1) and acontact groupoid

Theorem 12 Let 119872 be a generalized almost contact manifoldand 120578 a contact form There is a 1-1 correspondence between

(i) integrable Jacobi pair (119865 120587) on 119872 (ie (119865 120587) issatisfying (C1) integrable)

(ii) contact groupoids (Σ 120578) over119872

Proof Since (120578 119889120578) is a contact pair and (119865 120587) satisfies(C1) then (119865 120587) is an integrable Jacobi pair [18] FromDefinition 11 one sees that a contact groupoid is obtainedfrom an integrable Jacobi pair

The converse is clear

We now give the conditions for (C2) in terms of 119889120578 and120593

Lemma 13 Let 119872 be a generalized almost contact manifoldand 119889120578 a 2-form Given a nondegenerate bivector 120587 on119879119872lowast-120578 (ie 120587♯ = ((119889120578)

♯)minus1) and a map 120593 119879119872 rarr 119879119872

then 120587 and 120593 satisfy (C2) if and only if 119889120578 and 120593 commute

We now give a correspondence between generalizedcontact structures with nondegenerate 120587 and Hitchin pairs(119889120578 120593)

Proposition 14 There is a one to one correspondence betweengeneralized contact structures given by (11) with 120587 nonde-generate and Hitchin pairs (119889120578 120593) such that 119889120578(119883 119884) =

119889120578(120593119883 120593119884) In this correspondence 120587 is the inverse of 119889120578 and120579 is the twist of the Hitchin pair (119889120578 120593)

Proof Since (119889120578 120593) is Hitchin pair 119889120578 and (119889120578)120593are closed

By using the following equation (see [13])

119894119873120593(119883119884)

(119889120578) = 119894120593119883and119884+119883and120593119884

(119889(119889120578)120593) minus 119894120593119883and120593119884

(119889 (119889120578))

minus 119894119883and119884

(119889 (120593lowast119889120578))

(35)

we get

119894119873120593(119883119884)

(119889120578) = minus119894119883and119884

(119889 (120593lowast119889120578)) (36)

Since 120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894119865119889120578) we derive

(119889120578)♯(119873120593(119883 119884)) = minus119894

119883and119884(119889 (minus120579 + 120578 and (119894

119865119889120578))) (37)

Applying 120587♯ to (37) then we get

119873120593(119883 119884) = minus120587

♯(119894119883and119884

(119889 (minus120579 + 120578 and (119894119865119889120578))))

119873120593(119883 119884) = 120587

♯(119894119883and119884

(119889120579) minus 120587♯(119894119883and119884

119889120578 and (119894119865119889120578)))

119873120593(119883 119884) = 120587

♯(119894(119883and119884)

119889120579) minus 119889120578 (119883 119884) 119865

(38)

By assumption we have

119889120578 (119883 119884) = 119889120578 (120593119883 120593119884) (39)

Putting this equation into (38) we obtain

119873120593(119883 119884) = 120587

♯(119894(119883and119884)

119889120579) minus 119889120578 (120593119883 120593119884) 119865 (40)

which is the second equation of (C3) Now we show that120593lowast120579♯= 120579♯120593 From (26) we obtain

120593lowast120579♯= 120593lowast(minus(119889120578)

♯minus (120593lowast(119889120578))♯+ (120578 and (119894

119865(119889120578)))

♯) (41)

Hence we have

120593lowast120579♯= minus(119889120578)

♯120593 minus (120593

lowast(119889120578))♯120593 + (120578 and (119894

119865(119889120578)))

♯120593 (42)

From definition of twist we get

120593lowast120579♯= 120579♯120593 (43)

This equation is the first equation of (C4) Now wewill obtain

119889120579120593(119883 119884 119885)

= 119889120579 (120593119883 119884 119885) + 119889120579 (119883 120593119884 119885) + 119889120579 (119883 119884 120593119885)

(44)

which is second equation of (C4) Writing the equation as

119894119883and119884

(119889120579120593) = 119894120593119883and119884+119883and120593119884 (119889120579) + 120593

lowast(119894119883and119884 (119889120579)) (45)

and since 120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894

119865119889120578) then we should find

119894119883and119884

(119889 (minus(120593lowast(119889120578))120593+ (120578 and (119894

119865(119889120578)))

120593))

= 119894120593119883and119884+119883and120593119884

(119889 (minus120593lowast119889120578 + 120578 and (119894

119865119889120578)))

+ 120593lowast(119894119883and119884

(119889 (minus120593lowast119889120578 + 120578 and (119894

119865119889120578))))

(46)

A straightforward computation shows that

119894119883and119884

(119889 ((120593lowast(119889120578))120593)) = 119894

120593119883and119884+119883and120593119884(119889 (120593lowast119889120578))

+ 120593lowast(119894119883and119884

(119889 (120593lowast119889120578)))

(47)

6 Abstract and Applied Analysis

Using (35) then we get

119894119883and119884

(119889 ((120593lowast(119889120578))120593))

= 119894120593119883and119884+119883and120593119884

(119889 (120593lowast119889120578)) minus 119894

119873120593(119883119884)(119889120578)120593

(48)

Since 119894119873(119883119884)

((119889120578)120593) = 120593lowast119894

119873(119883119884)(119889120578) applying 119894

119883and119884119889(120593lowast119889120578)

= minus119894119873120593(119883119884)

(119889120578) to (48) we have

119894119883and119884

(119889 ((120593lowast(119889120578))120593))

= 119894120593119883and119884+119883and120593119884

(119889 (120593lowast119889120578)) + 120593

lowast(119894119883and119884

(119889 (120593lowast119889120578)))

(49)

In a similar way we can obtain (C5)The converse is clear from Lemmas 10 and 13

We note that similar to 2-forms given a Lie groupoid 119866a (1 1)-tensor 119869 119879119866 rarr 119879119866 is called multiplicative [13] iffor any (119892 ℎ) isin 119866 times 119866 and any V

119892isin 119879119892119866 119908ℎisin 119879ℎ119866 such that

(V119892 119908ℎ) is tangent to 119866 times 119866 at (119892 ℎ) so is (119869V

119892 119869119908ℎ) and

(119889119898)119892ℎ

(119869V119892 119869119908ℎ) = 119869 ((119889119898)

119892ℎ(V119892 119908ℎ)) (50)

Let (119872 120578) be a contact manifold Then it is easy to see thatthere is a one to one correspondence between (1 1)-tensors120593 commuting with 119889120578 and 2-forms on119872 On the other handit is easy to see that (C2) is equivalent to the fact that 120593lowast ∘ 119901119903

1

is an 119868119872 form on the Lie algebroid119879lowast119872oplusR associated Jacobistructure (119865 120587) Thus from the above discussion Lemma 13andTheorem 1 one can concludewith the following theorem

Theorem 15 Let119872 be a generalized almost contact manifoldLet (119865 120587) be an integrable Jacobi structure on 119872 and (Σ 120578)

a contact groupoid over 119872 Then there is a natural 1-1correspondence between

(i) (1 1)-tensors 120593 on119872 satisfying (C2)

(ii) multiplicative (1 1)-tensors 119868 on Σ with the propertythat (119868 119889120578) is a Hitchin pair

We recall the notion of generalized contact map betweengeneralized contact manifolds This notion is similar to thegeneralized holomorphic map given in [13]

Let (119872119894I119894) 119894 = 1 2 be two generalized contact

manifolds and let 120593119894 120587119894 120579119894be the components of I

119894in the

matrix representation (11) A map 119891 1198721

rarr 1198722is called

generalized contact if and only if 119891 maps 1205931into 120593

2 1198651into

1198652 and 120587

1into 120587

2 119891lowast1205792= 1205791and (119889119891) ∘ 120593

1= 1205932∘ (119889119891)

We now state and prove themain result of this paperThisresult gives equivalent assertions between the condition (C3)twist 120579 of (119889120578 119868) and contact maps for a contact groupoidover119872

Theorem 16 Let 119872 be a generalized almost contact manifoldand (Σ 120578 119868) an induced contact groupoid over 119872 with theinduced multiplicative (1 1)-tensor Assume that ((119865 120587) 119868)

satisfy (C1) (C2) with integrable (119865 120587) Then for a 120579 2-formon119872 the following assertions are equivalent

(i) (C3) is satisfied(ii) 119889120578 + 119868

lowast119889120578 minus 120578 and (119894119865119889120578) = 119904lowast120579 minus 119905lowast120579

(iii) (119905 119904) Σ rarr 119872 times 119872 is a generalized contact mapcondition of generalized contactmap on119872 is (119889119905)∘120593

1=

1205932∘ (119889119905) this condition on119872 is (119889119904) ∘ 120593

1= minus1205932∘ (119889119904)

Proof (i) hArr (ii) define 120601 = 120579minus119905lowast120579+119904lowast120579 such that 120579 = minus119889120578minus

119868lowast119889120578+120578and(119894119865119889120578) is the twist of (119889120578 119868) and119860 = ker(119889119904)|

119872 We

know fromTheorem 1 that closed multiplicative 2-form 120595 onΣ vanishes if and only if 119868119872 form 119906

120595= 0 that is120595(119883 120572) = 0

such that119883 isin 119879119872 120572 isin 119860 This case can be applied for formswith higher degree that is 3-form 120595 vanishes if and only if120595(119883 119884 120572) = 0

Since 119889120578 and (119889120578)119868

are closed from (35) we get119894119883and119884

(119889(119868lowast119889120578)) = minus119894119873119868(119883119884)

119889120578 Putting 120579 = minus119889120578 minus 119868lowast119889120578 + 120578 and

(119894119865119889120578) we obtain

119894119883and119884

119889 (minus120579 + 120578 and (119894119865119889120578)) = minus119894

119873119868(119883119884)119889120578 (51)

Since 119889120601 = 0 hArr 119889120601(119883 119884 120572) = 0 we have

119889120601 (119883 119884 120572) = 0 lArrrArr 119889120579 (119883 119884 120572) minus 119889 (119905lowast120579) (119883 119884 120572)

+ 119889 (119904lowast120579) (119883 119884 120572) = 0

(52)

On the other hand we obtain

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119889119905 (119883) 119889119905 (119884) 119889119905 (120572)) (53)

If we take 119889119905 = 120588 in (53) for 119860 we get

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119889119905 (119883) 119889119905 (119884) 120588 (120572)) (54)

On the other hand from [17] we know that

119868119889Σ= 119898 ∘ (119905 119868119889

Σ) (55)

Differentiating (55) we obtain

119883 = 119889119905 (119883) (56)

Using (56) in (54) we get

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119883 119884 120588 (120572)) (57)

In a similar way we see that

119889 (119904lowast120579) (119883 119884 120572) = 119889120579 (119889119904 (119883) 119889119904 (119884) 119889119904 (120572)) (58)

Since 120572 isin ker 119889119904 then 119889119904(120572) = 0 Hence 119889(119904lowast120579) = 0 Thuswe obtain

119889120579 (119883 119884 120572) = 119889120579 (119883 119884 120588 (120572)) (59)

Using (51) in (59) we derive

119889 (120578 and (119894119865119889120578)) (119883 119884 120572) + 119889120578 (119873

119868(119883 119884) 120572)

= 119889120579 (119883 119884 120588 (120572)) (60)

Abstract and Applied Analysis 7

On the other hand it is clear that 120601 = 0 hArr 120579 minus 119905lowast120579 + 119904lowast120579 = 0Thus we obtain

120579 (119883 120572) = 120579 (119883 120588 (120572)) (61)

Since 120579 = minus119889120578 minus 119868lowast119889120578 + 120578 and (119894119865119889120578) we get

minus 119889120578 (119883 120572) minus 119889120578 (119868119883 119868120572) + (120578 and (119894119865119889120578)) (119883 120572)

= 120579 (119883 120588 (120572)) (62)

Since Jacobi structure (119865 120587) is integrable it defines a Liealgebroid whose anchor map is 120588 = (120587 119865)

♯ Let us use (120587 119865)♯instead of 120588 in (60) and (62) then we get

119889 (120578 and (119894119865119889120578)) (119883 119884 120572) + 119889120578 (119873

119868(119883 119884) 120572)

= 119889120579 (119883 119884 120587♯(120572) + 119891119865)

(63)

minus 119889120578 (119883 120572) minus 119889120578 (119868119883 119868120572) + (120578 and (119894119865119889120578)) (119883 120572)

= 120579 (119883 120587♯(120572) + 119891119865)

(64)

Since 119889120578(120572119883) = 120572(119883) (119889120578)119868(120572 119883) = 120572(120593119883) from (63) we

have

minus 120572 (119889120578 (119883 119884) 119865) minus 120572 (119873120593(119883 119884))

= 119889120579 (119883 119884 120587♯(120572) + 119891119865)

= 119894119883and119884

119889120579 (120587♯(120572) + 119891119865)

= 119894119883and119884

119889120579 (120587♯(120572)) + 119894

119883and119884119889120579 (119891119865)

= 120587 (120572 119894119883and119884

119889120579)

= minus120572 (120587♯(119894119883and119884

119889120579))

(65)

that is 120572(119889120578(119883 119884)119865 + 119873120593(119883 119884)) = 120572(120587♯(119894

119883and119884119889120579))

Since the above equation holds for all nondegenerate 120572we get

119889120578 (119883 119884) 119865 + 119873120593 (119883 119884) = 120587

♯(119894119883and119884

119889120579) (66)

Then we arrive at

119889120578 (120593119883 120593119884) 119865 + 119873120593 (119883 119884) = 120587

♯(119894119883and119884

119889120579) (67)

On the other hand from (64) we obtain

120572 (119883) + 120572 (1205932119883) minus 120578 (119883) 120572 (119865) = 120587 (120572 119894

119883120579) + 119894119883120579 (119891119865)

= minus120572 (120587♯120579♯119883)

(68)

Thus we get

1205932+ 120587♯120579♯= minus119868119889 + 119865 ⊙ 120578 (69)

Then (i)hArr(ii) follows from (67) and (69)

(ii) hArr (iii) 119889120578+119868lowast119889120578minus120578and119894

119865119889120578 = 119904lowast120579minus119905lowast120579 says that (119905 119904)

is compatible with 2-form 119889120578 Also it is clear that (119905 119904) andbivectors are compatible because Σ is a contact groupoid Wewill check the compatibility of (119905 119904) and (1 1)-tensors Fromcompatibility condition of 119905 and 119904 we get 119889119905 ∘ 119868 = 120593 ∘ 119889119905 and119889119904 ∘ 119868 = minus120593 ∘ 119889119904

For all 120572 isin 119860 119881 isin 120594(Σ) and 119891 isin R we have

119889120578 (120572 119881) = 119889120578 (120572 119889119905119881) (70)

which is equivalent to

120572 (119881) = ⟨119906(119889120578)

(120572) 119889119905119881⟩ (71)

Since 119906(119889120578)

= 119868119889 and 119906((119889120578)119868)

= 120593lowast ∘ 1199011199031such that 119906

((119889120578)119868)(120572) =

120593lowast ∘ 1199011199031(120572 119891) = 120593lowast(120572) we get

⟨120572 120593 (119889119905 (119881))⟩ = 120572 (120593 (119889119905 (119881)))

= 120593lowast120572 (119889119905 (119881))

= ⟨119906((119889120578)119868)

120572 119881⟩

= 119889120578 (120572 119868119881)

= ⟨120572 119889119905 (119868119881)⟩

(72)

Since this equation holds for all 120572 isin 119860 119886(119889119905) = 119889119905(119868) Using119904 = 119905 ∘ 119894

119886 (119889119904 (119881)) = 119886119889 (119905 ∘ 119894) 119881

= 120593 (119889119905 (119889119894 (119881)))

= minus120593 (119889119905 (119881))

= minus119889119904 (119868119881)

(73)

which shows that 119886(119889119904) = minus119889119904(119868) Thus proof is completed

Conflict of Interests

The author declares that she has no conflict of interests

References

[1] C Ehresmann ldquoCategories topologiques et categoriesdifferentiablesrdquo in Colloque de Geometrie Differentielle GlobaleBruxelles Belgium pp 137ndash150 Librairie Universitaire 1959

[2] M Crainic and R L Fernandes ldquoIntegrability of Lie bracketsrdquoAnnals of Mathematics Second Series vol 157 no 2 pp 575ndash620 2003

[3] M Degeratu ldquoProjective limits of Lie groupoidsrdquo AnaleleUniversitatii din Oradea Fascicola Matematica vol 16 pp 183ndash188 2009

[4] M R Farhangdoost and T Nasirzade ldquoGeometrical categoriesof generalized Lie groups and Lie group-groupoidsrdquo IranianJournal of Science and Technology A Science vol 37 no 1 pp69ndash73 2013

[5] C-M Marle ldquoCalculus on Lie algebroids Lie groupoids andPoisson manifoldsrdquo Dissertationes Mathematicae (RozprawyMatematyczne) vol 457 p 57 2008

8 Abstract and Applied Analysis

[6] D S Zhong Z Chen and Z J Liu ldquoOn the existence of globalbisections of Lie groupoidsrdquo Acta Mathematica Sinica (EnglishSeries) vol 25 no 6 pp 1001ndash1014 2009

[7] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiablesCalcul differenetiel dans la categorie des groupoıdesinfinitesimauxrdquo Comptes Rendus de lrsquoAcademie des Sciences Avol 264 pp 245ndash248 1967

[8] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquoThe QuarterlyJournal of Mathematics vol 54 no 3 pp 281ndash308 2003

[9] M Gualtieri Generalized complex geometry [PhD thesis]University of Oxford 2003

[10] D Iglesias-Ponte and A Wade ldquoContact manifolds and gener-alized complex structuresrdquo Journal of Geometry and Physics vol53 no 3 pp 249ndash258 2005

[11] Y S Poon and A Wade ldquoGeneralized contact structuresrdquoJournal of the London Mathematical Society Second Series vol83 no 2 pp 333ndash352 2011

[12] A Wade ldquoLocal structure of generalized contact manifoldsrdquoDifferential Geometry and its Applications vol 30 no 1 pp 124ndash135 2012

[13] M Crainic ldquoGeneralized complex structures and Lie bracketsrdquoBulletin of the Brazilian Mathematical Society New SeriesBoletim da Sociedade Brasileira de Matematica vol 42 no 4pp 559ndash578 2011

[14] M Crainic and C Zhu ldquoIntegrability of Jacobi structuresrdquoAnnales de lrsquoinstitut Fourier vol 57 no 4 pp 1181ndash1216 2007

[15] A A Kirillov ldquoLocal Lie algebrasrdquo Russian MathematicalSurveys vol 31 pp 55ndash75 1976

[16] M Crainic and R L Fernandes ldquoLectures on integrability of Liebracketsrdquo in Lectures on Poisson Geometry vol 17 of Geometryand Topology Monographs pp 1ndash107 Geometry amp TopologyPublications Coventry UK 2011

[17] H BursztynM Crainic AWeinstein and C Zhu ldquoIntegrationof twisted Dirac bracketsrdquo Duke Mathematical Journal vol 123no 3 pp 549ndash607 2004

[18] J Janyska and M Modugno ldquoGeneralized geometrical struc-tures of odd dimensional manifoldsrdquo Journal de MathematiquesPures et Appliquees Neuvieme Serie vol 91 no 2 pp 211ndash2322009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

(C2)

120593120587♯= 120587♯120593lowast

120593lowast[120572 120573]

120587= L120587♯120572120593lowast120573 minusL

120587♯120573120593lowast120572 minus 119889120587 (120593

lowast120572 120573)

(19)

(C3)

1205932+ 120587♯120579♯= minus119868119889 + 119865 ⊙ 120578

119873120593 (119883 119884) + 119889120578 (120593119883 120593119884) 119865 = 120587

♯(119894119883and119884

119889120579)

(20)

(C4)

120593lowast120579♯= 120579♯120593

119889120579120593 (119883 119884 119885)

= 119889120579 (120593119883 119884 119885) + 119889120579 (119883 120593119884 119885) + 119889120579 (119883 119884 120593119885)

(21)

(C5)

L119865120593 = 0 L

119865120579 = 0 (22)

where

[120572 120573]120587= L120587♯120572120573 minusL

120587♯120573120572 minus 119889120587 (120572 120573)

120579120593 (119883 119884) = 120579 (120593119883 119884)

(23)

We note that if (11) is a generalized contact structure then

I = [120593 minus120587♯

minus120579♯

minus120593lowast] (24)

is also a generalized contact structureI is called the oppositeofI In this paper we denote a generalized contact manifoldendowed withI by119872

As an analogue of aHitchin pair on a generalized complexmanifold a Hitchin pair on a generalized almost contactmanifold 119872 is a pair (119889120578 120593) consisting of a contact form120578 and a (1 1)-tensor 120593 with the property that 119889120578 and 120593

commute (ie 119889120578(119883 120593119884) = 119889120578(120593119883 119884)) We note that sincea generalized almost contact structure is equivalent to ageneralized almost complex structure on119872timesR the bivectorfield 120587 of the generalized almost contact structure is notnondegenerate in general But we emphasize that we areputting this condition for restricted case

Lemma 8 Let119872 be a generalized almost contact manifold If120587 is a nondegenerate bivector field on 119879119872

lowast-119878119901119886119899120578 119889120578 is theinverse 2-form (defined by (119889120578)

♯= (120587♯)

minus1) and120587 satisfies (20)then 120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894

119865119889120578) if and only if 119889120578(1205932119883119884) =

120593lowast119889120578(119883 119884)

Proof For 119883 isin 120594(119872) we apply (119889120578)♯to (20) and using the

dual structure 120593lowast we have

(119889120578)♯1205932(119883)

= minus(119889120578)♯(119883) minus (119889120578)

♯(120587♯120579♯ (119883)) + (119889120578)

♯(120578 (119883) 119865)

119889120578 (1205932119883119884) = minus119889120578 (119883 119884) minus 120579

♯(119883) (119884) + 119889120578 (120578 (119883) 119865 119884)

(25)

We obtain

120593lowast119889120578 (119883 119884) = minus119889120578 (119883 119884) minus 120579 (119883 119884) + 120578 (119883) 119889120578 (119865 119884)

(26)

Since (26) holds for all119883 and 119884 we get

120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894

119865119889120578) (27)

In a similar way one can get the converse

From now on when we mention a nondegenerate bivec-tor field 120587 we mean it is nondegenerate on 119879119872lowast-Span120578We note that if 119889120578 is the inverse 2-form of 120587 nondegenerate120587 on 119879119872

lowast-Span120578 implies that 119889120578 is also nondegenerate on119879119872-Span119865

We say that 2-form 120579 is the twist of Hitchin pair (119889120578 120593)Note that in this case 120593 is neither an almost contact structurenor torsion(119873

120593) free

Lemma 9 Let (119872 120578 120593 119865) be an almost contact manifold 119889120578and 120593 commute if and only if 119889120578 + 120593lowast119889120578 = 120578 and (119894

119865119889120578)

Proof We will only prove the sufficient condition We have

120593lowast119889120578 (119883 119884) + 119889120578 (119883 119884) minus 120578 and (119894

119865119889120578) (119883 119884) = 0 (28)

Since 120593lowast is dual contact structure we get

119889120578 (120593119883 120593119884) + 119889120578 (119883 119884) minus 120578 (119883) (119894119865119889120578) (119884) = 0 (29)

Substituting 120593119883 by119883 and using contact structure property

119889120578 (minus119883 + 120578 (119883) 119865 120593119884) + 119889120578 (120593119883 119884) = 0 (30)

Hence we obtain

minus119889120578 (119883 120593119884) + 119889120578 (120593119883 119884) = 0 (31)

which shows that 119889120578 and 120593 commute The converse is clear

Next we see that (C1) is satisfied automatically when onechooses 119889120578 as the 2-form which is the inverse of 120587 defined by(119889120578)♯= (120587♯)

minus1

Lemma 10 Let120587 be a nondegenerate bivector on a generalizedalmost contact manifold119872 and 119889120578 the inverse 2-form (definedby (119889120578)

♯= (120587♯)

minus1) Then 120587 satisfies (C1)

Proof Since 119889120578 is a closed form it is obvious due to [13Lemma 27]

Abstract and Applied Analysis 5

Definition 11 (see [14]) The Lie algebroid of the Jacobimanifold (119872 120587 119865) is 119879lowast119872 oplus R with the anchor 120588 119879lowast119872 oplus

R rarr 119879119872 given by

120588 (120596 120582) = (120587 119865)♯(120596 120582) = 120587

♯(120596) + 120582119865 (32)

and the bracket

[(120596 0) (120578 0)] = ([120596 120578]120587 0) minus (119894

119865(120596 and 120578) 120587 (120596 120578))

[(0 1) (120596 0)] = (L(119865)

120596 0) (33)

The associated groupoid

Σ (119872) = 119866 (119879lowast119872 oplusR) (34)

is called contact groupoid of the Jacobi manifold 119872 We saythat 119872 is integrable as a Jacobi manifold if the associatedalgebroid 119879lowast119872 oplus R is integrable (or equivalently if Σ(119872) issmooth)

Thus we have the following result which shows that thereis a close relationship between the condition (C1) and acontact groupoid

Theorem 12 Let 119872 be a generalized almost contact manifoldand 120578 a contact form There is a 1-1 correspondence between

(i) integrable Jacobi pair (119865 120587) on 119872 (ie (119865 120587) issatisfying (C1) integrable)

(ii) contact groupoids (Σ 120578) over119872

Proof Since (120578 119889120578) is a contact pair and (119865 120587) satisfies(C1) then (119865 120587) is an integrable Jacobi pair [18] FromDefinition 11 one sees that a contact groupoid is obtainedfrom an integrable Jacobi pair

The converse is clear

We now give the conditions for (C2) in terms of 119889120578 and120593

Lemma 13 Let 119872 be a generalized almost contact manifoldand 119889120578 a 2-form Given a nondegenerate bivector 120587 on119879119872lowast-120578 (ie 120587♯ = ((119889120578)

♯)minus1) and a map 120593 119879119872 rarr 119879119872

then 120587 and 120593 satisfy (C2) if and only if 119889120578 and 120593 commute

We now give a correspondence between generalizedcontact structures with nondegenerate 120587 and Hitchin pairs(119889120578 120593)

Proposition 14 There is a one to one correspondence betweengeneralized contact structures given by (11) with 120587 nonde-generate and Hitchin pairs (119889120578 120593) such that 119889120578(119883 119884) =

119889120578(120593119883 120593119884) In this correspondence 120587 is the inverse of 119889120578 and120579 is the twist of the Hitchin pair (119889120578 120593)

Proof Since (119889120578 120593) is Hitchin pair 119889120578 and (119889120578)120593are closed

By using the following equation (see [13])

119894119873120593(119883119884)

(119889120578) = 119894120593119883and119884+119883and120593119884

(119889(119889120578)120593) minus 119894120593119883and120593119884

(119889 (119889120578))

minus 119894119883and119884

(119889 (120593lowast119889120578))

(35)

we get

119894119873120593(119883119884)

(119889120578) = minus119894119883and119884

(119889 (120593lowast119889120578)) (36)

Since 120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894119865119889120578) we derive

(119889120578)♯(119873120593(119883 119884)) = minus119894

119883and119884(119889 (minus120579 + 120578 and (119894

119865119889120578))) (37)

Applying 120587♯ to (37) then we get

119873120593(119883 119884) = minus120587

♯(119894119883and119884

(119889 (minus120579 + 120578 and (119894119865119889120578))))

119873120593(119883 119884) = 120587

♯(119894119883and119884

(119889120579) minus 120587♯(119894119883and119884

119889120578 and (119894119865119889120578)))

119873120593(119883 119884) = 120587

♯(119894(119883and119884)

119889120579) minus 119889120578 (119883 119884) 119865

(38)

By assumption we have

119889120578 (119883 119884) = 119889120578 (120593119883 120593119884) (39)

Putting this equation into (38) we obtain

119873120593(119883 119884) = 120587

♯(119894(119883and119884)

119889120579) minus 119889120578 (120593119883 120593119884) 119865 (40)

which is the second equation of (C3) Now we show that120593lowast120579♯= 120579♯120593 From (26) we obtain

120593lowast120579♯= 120593lowast(minus(119889120578)

♯minus (120593lowast(119889120578))♯+ (120578 and (119894

119865(119889120578)))

♯) (41)

Hence we have

120593lowast120579♯= minus(119889120578)

♯120593 minus (120593

lowast(119889120578))♯120593 + (120578 and (119894

119865(119889120578)))

♯120593 (42)

From definition of twist we get

120593lowast120579♯= 120579♯120593 (43)

This equation is the first equation of (C4) Now wewill obtain

119889120579120593(119883 119884 119885)

= 119889120579 (120593119883 119884 119885) + 119889120579 (119883 120593119884 119885) + 119889120579 (119883 119884 120593119885)

(44)

which is second equation of (C4) Writing the equation as

119894119883and119884

(119889120579120593) = 119894120593119883and119884+119883and120593119884 (119889120579) + 120593

lowast(119894119883and119884 (119889120579)) (45)

and since 120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894

119865119889120578) then we should find

119894119883and119884

(119889 (minus(120593lowast(119889120578))120593+ (120578 and (119894

119865(119889120578)))

120593))

= 119894120593119883and119884+119883and120593119884

(119889 (minus120593lowast119889120578 + 120578 and (119894

119865119889120578)))

+ 120593lowast(119894119883and119884

(119889 (minus120593lowast119889120578 + 120578 and (119894

119865119889120578))))

(46)

A straightforward computation shows that

119894119883and119884

(119889 ((120593lowast(119889120578))120593)) = 119894

120593119883and119884+119883and120593119884(119889 (120593lowast119889120578))

+ 120593lowast(119894119883and119884

(119889 (120593lowast119889120578)))

(47)

6 Abstract and Applied Analysis

Using (35) then we get

119894119883and119884

(119889 ((120593lowast(119889120578))120593))

= 119894120593119883and119884+119883and120593119884

(119889 (120593lowast119889120578)) minus 119894

119873120593(119883119884)(119889120578)120593

(48)

Since 119894119873(119883119884)

((119889120578)120593) = 120593lowast119894

119873(119883119884)(119889120578) applying 119894

119883and119884119889(120593lowast119889120578)

= minus119894119873120593(119883119884)

(119889120578) to (48) we have

119894119883and119884

(119889 ((120593lowast(119889120578))120593))

= 119894120593119883and119884+119883and120593119884

(119889 (120593lowast119889120578)) + 120593

lowast(119894119883and119884

(119889 (120593lowast119889120578)))

(49)

In a similar way we can obtain (C5)The converse is clear from Lemmas 10 and 13

We note that similar to 2-forms given a Lie groupoid 119866a (1 1)-tensor 119869 119879119866 rarr 119879119866 is called multiplicative [13] iffor any (119892 ℎ) isin 119866 times 119866 and any V

119892isin 119879119892119866 119908ℎisin 119879ℎ119866 such that

(V119892 119908ℎ) is tangent to 119866 times 119866 at (119892 ℎ) so is (119869V

119892 119869119908ℎ) and

(119889119898)119892ℎ

(119869V119892 119869119908ℎ) = 119869 ((119889119898)

119892ℎ(V119892 119908ℎ)) (50)

Let (119872 120578) be a contact manifold Then it is easy to see thatthere is a one to one correspondence between (1 1)-tensors120593 commuting with 119889120578 and 2-forms on119872 On the other handit is easy to see that (C2) is equivalent to the fact that 120593lowast ∘ 119901119903

1

is an 119868119872 form on the Lie algebroid119879lowast119872oplusR associated Jacobistructure (119865 120587) Thus from the above discussion Lemma 13andTheorem 1 one can concludewith the following theorem

Theorem 15 Let119872 be a generalized almost contact manifoldLet (119865 120587) be an integrable Jacobi structure on 119872 and (Σ 120578)

a contact groupoid over 119872 Then there is a natural 1-1correspondence between

(i) (1 1)-tensors 120593 on119872 satisfying (C2)

(ii) multiplicative (1 1)-tensors 119868 on Σ with the propertythat (119868 119889120578) is a Hitchin pair

We recall the notion of generalized contact map betweengeneralized contact manifolds This notion is similar to thegeneralized holomorphic map given in [13]

Let (119872119894I119894) 119894 = 1 2 be two generalized contact

manifolds and let 120593119894 120587119894 120579119894be the components of I

119894in the

matrix representation (11) A map 119891 1198721

rarr 1198722is called

generalized contact if and only if 119891 maps 1205931into 120593

2 1198651into

1198652 and 120587

1into 120587

2 119891lowast1205792= 1205791and (119889119891) ∘ 120593

1= 1205932∘ (119889119891)

We now state and prove themain result of this paperThisresult gives equivalent assertions between the condition (C3)twist 120579 of (119889120578 119868) and contact maps for a contact groupoidover119872

Theorem 16 Let 119872 be a generalized almost contact manifoldand (Σ 120578 119868) an induced contact groupoid over 119872 with theinduced multiplicative (1 1)-tensor Assume that ((119865 120587) 119868)

satisfy (C1) (C2) with integrable (119865 120587) Then for a 120579 2-formon119872 the following assertions are equivalent

(i) (C3) is satisfied(ii) 119889120578 + 119868

lowast119889120578 minus 120578 and (119894119865119889120578) = 119904lowast120579 minus 119905lowast120579

(iii) (119905 119904) Σ rarr 119872 times 119872 is a generalized contact mapcondition of generalized contactmap on119872 is (119889119905)∘120593

1=

1205932∘ (119889119905) this condition on119872 is (119889119904) ∘ 120593

1= minus1205932∘ (119889119904)

Proof (i) hArr (ii) define 120601 = 120579minus119905lowast120579+119904lowast120579 such that 120579 = minus119889120578minus

119868lowast119889120578+120578and(119894119865119889120578) is the twist of (119889120578 119868) and119860 = ker(119889119904)|

119872 We

know fromTheorem 1 that closed multiplicative 2-form 120595 onΣ vanishes if and only if 119868119872 form 119906

120595= 0 that is120595(119883 120572) = 0

such that119883 isin 119879119872 120572 isin 119860 This case can be applied for formswith higher degree that is 3-form 120595 vanishes if and only if120595(119883 119884 120572) = 0

Since 119889120578 and (119889120578)119868

are closed from (35) we get119894119883and119884

(119889(119868lowast119889120578)) = minus119894119873119868(119883119884)

119889120578 Putting 120579 = minus119889120578 minus 119868lowast119889120578 + 120578 and

(119894119865119889120578) we obtain

119894119883and119884

119889 (minus120579 + 120578 and (119894119865119889120578)) = minus119894

119873119868(119883119884)119889120578 (51)

Since 119889120601 = 0 hArr 119889120601(119883 119884 120572) = 0 we have

119889120601 (119883 119884 120572) = 0 lArrrArr 119889120579 (119883 119884 120572) minus 119889 (119905lowast120579) (119883 119884 120572)

+ 119889 (119904lowast120579) (119883 119884 120572) = 0

(52)

On the other hand we obtain

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119889119905 (119883) 119889119905 (119884) 119889119905 (120572)) (53)

If we take 119889119905 = 120588 in (53) for 119860 we get

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119889119905 (119883) 119889119905 (119884) 120588 (120572)) (54)

On the other hand from [17] we know that

119868119889Σ= 119898 ∘ (119905 119868119889

Σ) (55)

Differentiating (55) we obtain

119883 = 119889119905 (119883) (56)

Using (56) in (54) we get

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119883 119884 120588 (120572)) (57)

In a similar way we see that

119889 (119904lowast120579) (119883 119884 120572) = 119889120579 (119889119904 (119883) 119889119904 (119884) 119889119904 (120572)) (58)

Since 120572 isin ker 119889119904 then 119889119904(120572) = 0 Hence 119889(119904lowast120579) = 0 Thuswe obtain

119889120579 (119883 119884 120572) = 119889120579 (119883 119884 120588 (120572)) (59)

Using (51) in (59) we derive

119889 (120578 and (119894119865119889120578)) (119883 119884 120572) + 119889120578 (119873

119868(119883 119884) 120572)

= 119889120579 (119883 119884 120588 (120572)) (60)

Abstract and Applied Analysis 7

On the other hand it is clear that 120601 = 0 hArr 120579 minus 119905lowast120579 + 119904lowast120579 = 0Thus we obtain

120579 (119883 120572) = 120579 (119883 120588 (120572)) (61)

Since 120579 = minus119889120578 minus 119868lowast119889120578 + 120578 and (119894119865119889120578) we get

minus 119889120578 (119883 120572) minus 119889120578 (119868119883 119868120572) + (120578 and (119894119865119889120578)) (119883 120572)

= 120579 (119883 120588 (120572)) (62)

Since Jacobi structure (119865 120587) is integrable it defines a Liealgebroid whose anchor map is 120588 = (120587 119865)

♯ Let us use (120587 119865)♯instead of 120588 in (60) and (62) then we get

119889 (120578 and (119894119865119889120578)) (119883 119884 120572) + 119889120578 (119873

119868(119883 119884) 120572)

= 119889120579 (119883 119884 120587♯(120572) + 119891119865)

(63)

minus 119889120578 (119883 120572) minus 119889120578 (119868119883 119868120572) + (120578 and (119894119865119889120578)) (119883 120572)

= 120579 (119883 120587♯(120572) + 119891119865)

(64)

Since 119889120578(120572119883) = 120572(119883) (119889120578)119868(120572 119883) = 120572(120593119883) from (63) we

have

minus 120572 (119889120578 (119883 119884) 119865) minus 120572 (119873120593(119883 119884))

= 119889120579 (119883 119884 120587♯(120572) + 119891119865)

= 119894119883and119884

119889120579 (120587♯(120572) + 119891119865)

= 119894119883and119884

119889120579 (120587♯(120572)) + 119894

119883and119884119889120579 (119891119865)

= 120587 (120572 119894119883and119884

119889120579)

= minus120572 (120587♯(119894119883and119884

119889120579))

(65)

that is 120572(119889120578(119883 119884)119865 + 119873120593(119883 119884)) = 120572(120587♯(119894

119883and119884119889120579))

Since the above equation holds for all nondegenerate 120572we get

119889120578 (119883 119884) 119865 + 119873120593 (119883 119884) = 120587

♯(119894119883and119884

119889120579) (66)

Then we arrive at

119889120578 (120593119883 120593119884) 119865 + 119873120593 (119883 119884) = 120587

♯(119894119883and119884

119889120579) (67)

On the other hand from (64) we obtain

120572 (119883) + 120572 (1205932119883) minus 120578 (119883) 120572 (119865) = 120587 (120572 119894

119883120579) + 119894119883120579 (119891119865)

= minus120572 (120587♯120579♯119883)

(68)

Thus we get

1205932+ 120587♯120579♯= minus119868119889 + 119865 ⊙ 120578 (69)

Then (i)hArr(ii) follows from (67) and (69)

(ii) hArr (iii) 119889120578+119868lowast119889120578minus120578and119894

119865119889120578 = 119904lowast120579minus119905lowast120579 says that (119905 119904)

is compatible with 2-form 119889120578 Also it is clear that (119905 119904) andbivectors are compatible because Σ is a contact groupoid Wewill check the compatibility of (119905 119904) and (1 1)-tensors Fromcompatibility condition of 119905 and 119904 we get 119889119905 ∘ 119868 = 120593 ∘ 119889119905 and119889119904 ∘ 119868 = minus120593 ∘ 119889119904

For all 120572 isin 119860 119881 isin 120594(Σ) and 119891 isin R we have

119889120578 (120572 119881) = 119889120578 (120572 119889119905119881) (70)

which is equivalent to

120572 (119881) = ⟨119906(119889120578)

(120572) 119889119905119881⟩ (71)

Since 119906(119889120578)

= 119868119889 and 119906((119889120578)119868)

= 120593lowast ∘ 1199011199031such that 119906

((119889120578)119868)(120572) =

120593lowast ∘ 1199011199031(120572 119891) = 120593lowast(120572) we get

⟨120572 120593 (119889119905 (119881))⟩ = 120572 (120593 (119889119905 (119881)))

= 120593lowast120572 (119889119905 (119881))

= ⟨119906((119889120578)119868)

120572 119881⟩

= 119889120578 (120572 119868119881)

= ⟨120572 119889119905 (119868119881)⟩

(72)

Since this equation holds for all 120572 isin 119860 119886(119889119905) = 119889119905(119868) Using119904 = 119905 ∘ 119894

119886 (119889119904 (119881)) = 119886119889 (119905 ∘ 119894) 119881

= 120593 (119889119905 (119889119894 (119881)))

= minus120593 (119889119905 (119881))

= minus119889119904 (119868119881)

(73)

which shows that 119886(119889119904) = minus119889119904(119868) Thus proof is completed

Conflict of Interests

The author declares that she has no conflict of interests

References

[1] C Ehresmann ldquoCategories topologiques et categoriesdifferentiablesrdquo in Colloque de Geometrie Differentielle GlobaleBruxelles Belgium pp 137ndash150 Librairie Universitaire 1959

[2] M Crainic and R L Fernandes ldquoIntegrability of Lie bracketsrdquoAnnals of Mathematics Second Series vol 157 no 2 pp 575ndash620 2003

[3] M Degeratu ldquoProjective limits of Lie groupoidsrdquo AnaleleUniversitatii din Oradea Fascicola Matematica vol 16 pp 183ndash188 2009

[4] M R Farhangdoost and T Nasirzade ldquoGeometrical categoriesof generalized Lie groups and Lie group-groupoidsrdquo IranianJournal of Science and Technology A Science vol 37 no 1 pp69ndash73 2013

[5] C-M Marle ldquoCalculus on Lie algebroids Lie groupoids andPoisson manifoldsrdquo Dissertationes Mathematicae (RozprawyMatematyczne) vol 457 p 57 2008

8 Abstract and Applied Analysis

[6] D S Zhong Z Chen and Z J Liu ldquoOn the existence of globalbisections of Lie groupoidsrdquo Acta Mathematica Sinica (EnglishSeries) vol 25 no 6 pp 1001ndash1014 2009

[7] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiablesCalcul differenetiel dans la categorie des groupoıdesinfinitesimauxrdquo Comptes Rendus de lrsquoAcademie des Sciences Avol 264 pp 245ndash248 1967

[8] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquoThe QuarterlyJournal of Mathematics vol 54 no 3 pp 281ndash308 2003

[9] M Gualtieri Generalized complex geometry [PhD thesis]University of Oxford 2003

[10] D Iglesias-Ponte and A Wade ldquoContact manifolds and gener-alized complex structuresrdquo Journal of Geometry and Physics vol53 no 3 pp 249ndash258 2005

[11] Y S Poon and A Wade ldquoGeneralized contact structuresrdquoJournal of the London Mathematical Society Second Series vol83 no 2 pp 333ndash352 2011

[12] A Wade ldquoLocal structure of generalized contact manifoldsrdquoDifferential Geometry and its Applications vol 30 no 1 pp 124ndash135 2012

[13] M Crainic ldquoGeneralized complex structures and Lie bracketsrdquoBulletin of the Brazilian Mathematical Society New SeriesBoletim da Sociedade Brasileira de Matematica vol 42 no 4pp 559ndash578 2011

[14] M Crainic and C Zhu ldquoIntegrability of Jacobi structuresrdquoAnnales de lrsquoinstitut Fourier vol 57 no 4 pp 1181ndash1216 2007

[15] A A Kirillov ldquoLocal Lie algebrasrdquo Russian MathematicalSurveys vol 31 pp 55ndash75 1976

[16] M Crainic and R L Fernandes ldquoLectures on integrability of Liebracketsrdquo in Lectures on Poisson Geometry vol 17 of Geometryand Topology Monographs pp 1ndash107 Geometry amp TopologyPublications Coventry UK 2011

[17] H BursztynM Crainic AWeinstein and C Zhu ldquoIntegrationof twisted Dirac bracketsrdquo Duke Mathematical Journal vol 123no 3 pp 549ndash607 2004

[18] J Janyska and M Modugno ldquoGeneralized geometrical struc-tures of odd dimensional manifoldsrdquo Journal de MathematiquesPures et Appliquees Neuvieme Serie vol 91 no 2 pp 211ndash2322009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Definition 11 (see [14]) The Lie algebroid of the Jacobimanifold (119872 120587 119865) is 119879lowast119872 oplus R with the anchor 120588 119879lowast119872 oplus

R rarr 119879119872 given by

120588 (120596 120582) = (120587 119865)♯(120596 120582) = 120587

♯(120596) + 120582119865 (32)

and the bracket

[(120596 0) (120578 0)] = ([120596 120578]120587 0) minus (119894

119865(120596 and 120578) 120587 (120596 120578))

[(0 1) (120596 0)] = (L(119865)

120596 0) (33)

The associated groupoid

Σ (119872) = 119866 (119879lowast119872 oplusR) (34)

is called contact groupoid of the Jacobi manifold 119872 We saythat 119872 is integrable as a Jacobi manifold if the associatedalgebroid 119879lowast119872 oplus R is integrable (or equivalently if Σ(119872) issmooth)

Thus we have the following result which shows that thereis a close relationship between the condition (C1) and acontact groupoid

Theorem 12 Let 119872 be a generalized almost contact manifoldand 120578 a contact form There is a 1-1 correspondence between

(i) integrable Jacobi pair (119865 120587) on 119872 (ie (119865 120587) issatisfying (C1) integrable)

(ii) contact groupoids (Σ 120578) over119872

Proof Since (120578 119889120578) is a contact pair and (119865 120587) satisfies(C1) then (119865 120587) is an integrable Jacobi pair [18] FromDefinition 11 one sees that a contact groupoid is obtainedfrom an integrable Jacobi pair

The converse is clear

We now give the conditions for (C2) in terms of 119889120578 and120593

Lemma 13 Let 119872 be a generalized almost contact manifoldand 119889120578 a 2-form Given a nondegenerate bivector 120587 on119879119872lowast-120578 (ie 120587♯ = ((119889120578)

♯)minus1) and a map 120593 119879119872 rarr 119879119872

then 120587 and 120593 satisfy (C2) if and only if 119889120578 and 120593 commute

We now give a correspondence between generalizedcontact structures with nondegenerate 120587 and Hitchin pairs(119889120578 120593)

Proposition 14 There is a one to one correspondence betweengeneralized contact structures given by (11) with 120587 nonde-generate and Hitchin pairs (119889120578 120593) such that 119889120578(119883 119884) =

119889120578(120593119883 120593119884) In this correspondence 120587 is the inverse of 119889120578 and120579 is the twist of the Hitchin pair (119889120578 120593)

Proof Since (119889120578 120593) is Hitchin pair 119889120578 and (119889120578)120593are closed

By using the following equation (see [13])

119894119873120593(119883119884)

(119889120578) = 119894120593119883and119884+119883and120593119884

(119889(119889120578)120593) minus 119894120593119883and120593119884

(119889 (119889120578))

minus 119894119883and119884

(119889 (120593lowast119889120578))

(35)

we get

119894119873120593(119883119884)

(119889120578) = minus119894119883and119884

(119889 (120593lowast119889120578)) (36)

Since 120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894119865119889120578) we derive

(119889120578)♯(119873120593(119883 119884)) = minus119894

119883and119884(119889 (minus120579 + 120578 and (119894

119865119889120578))) (37)

Applying 120587♯ to (37) then we get

119873120593(119883 119884) = minus120587

♯(119894119883and119884

(119889 (minus120579 + 120578 and (119894119865119889120578))))

119873120593(119883 119884) = 120587

♯(119894119883and119884

(119889120579) minus 120587♯(119894119883and119884

119889120578 and (119894119865119889120578)))

119873120593(119883 119884) = 120587

♯(119894(119883and119884)

119889120579) minus 119889120578 (119883 119884) 119865

(38)

By assumption we have

119889120578 (119883 119884) = 119889120578 (120593119883 120593119884) (39)

Putting this equation into (38) we obtain

119873120593(119883 119884) = 120587

♯(119894(119883and119884)

119889120579) minus 119889120578 (120593119883 120593119884) 119865 (40)

which is the second equation of (C3) Now we show that120593lowast120579♯= 120579♯120593 From (26) we obtain

120593lowast120579♯= 120593lowast(minus(119889120578)

♯minus (120593lowast(119889120578))♯+ (120578 and (119894

119865(119889120578)))

♯) (41)

Hence we have

120593lowast120579♯= minus(119889120578)

♯120593 minus (120593

lowast(119889120578))♯120593 + (120578 and (119894

119865(119889120578)))

♯120593 (42)

From definition of twist we get

120593lowast120579♯= 120579♯120593 (43)

This equation is the first equation of (C4) Now wewill obtain

119889120579120593(119883 119884 119885)

= 119889120579 (120593119883 119884 119885) + 119889120579 (119883 120593119884 119885) + 119889120579 (119883 119884 120593119885)

(44)

which is second equation of (C4) Writing the equation as

119894119883and119884

(119889120579120593) = 119894120593119883and119884+119883and120593119884 (119889120579) + 120593

lowast(119894119883and119884 (119889120579)) (45)

and since 120579 = minus119889120578 minus 120593lowast119889120578 + 120578 and (119894

119865119889120578) then we should find

119894119883and119884

(119889 (minus(120593lowast(119889120578))120593+ (120578 and (119894

119865(119889120578)))

120593))

= 119894120593119883and119884+119883and120593119884

(119889 (minus120593lowast119889120578 + 120578 and (119894

119865119889120578)))

+ 120593lowast(119894119883and119884

(119889 (minus120593lowast119889120578 + 120578 and (119894

119865119889120578))))

(46)

A straightforward computation shows that

119894119883and119884

(119889 ((120593lowast(119889120578))120593)) = 119894

120593119883and119884+119883and120593119884(119889 (120593lowast119889120578))

+ 120593lowast(119894119883and119884

(119889 (120593lowast119889120578)))

(47)

6 Abstract and Applied Analysis

Using (35) then we get

119894119883and119884

(119889 ((120593lowast(119889120578))120593))

= 119894120593119883and119884+119883and120593119884

(119889 (120593lowast119889120578)) minus 119894

119873120593(119883119884)(119889120578)120593

(48)

Since 119894119873(119883119884)

((119889120578)120593) = 120593lowast119894

119873(119883119884)(119889120578) applying 119894

119883and119884119889(120593lowast119889120578)

= minus119894119873120593(119883119884)

(119889120578) to (48) we have

119894119883and119884

(119889 ((120593lowast(119889120578))120593))

= 119894120593119883and119884+119883and120593119884

(119889 (120593lowast119889120578)) + 120593

lowast(119894119883and119884

(119889 (120593lowast119889120578)))

(49)

In a similar way we can obtain (C5)The converse is clear from Lemmas 10 and 13

We note that similar to 2-forms given a Lie groupoid 119866a (1 1)-tensor 119869 119879119866 rarr 119879119866 is called multiplicative [13] iffor any (119892 ℎ) isin 119866 times 119866 and any V

119892isin 119879119892119866 119908ℎisin 119879ℎ119866 such that

(V119892 119908ℎ) is tangent to 119866 times 119866 at (119892 ℎ) so is (119869V

119892 119869119908ℎ) and

(119889119898)119892ℎ

(119869V119892 119869119908ℎ) = 119869 ((119889119898)

119892ℎ(V119892 119908ℎ)) (50)

Let (119872 120578) be a contact manifold Then it is easy to see thatthere is a one to one correspondence between (1 1)-tensors120593 commuting with 119889120578 and 2-forms on119872 On the other handit is easy to see that (C2) is equivalent to the fact that 120593lowast ∘ 119901119903

1

is an 119868119872 form on the Lie algebroid119879lowast119872oplusR associated Jacobistructure (119865 120587) Thus from the above discussion Lemma 13andTheorem 1 one can concludewith the following theorem

Theorem 15 Let119872 be a generalized almost contact manifoldLet (119865 120587) be an integrable Jacobi structure on 119872 and (Σ 120578)

a contact groupoid over 119872 Then there is a natural 1-1correspondence between

(i) (1 1)-tensors 120593 on119872 satisfying (C2)

(ii) multiplicative (1 1)-tensors 119868 on Σ with the propertythat (119868 119889120578) is a Hitchin pair

We recall the notion of generalized contact map betweengeneralized contact manifolds This notion is similar to thegeneralized holomorphic map given in [13]

Let (119872119894I119894) 119894 = 1 2 be two generalized contact

manifolds and let 120593119894 120587119894 120579119894be the components of I

119894in the

matrix representation (11) A map 119891 1198721

rarr 1198722is called

generalized contact if and only if 119891 maps 1205931into 120593

2 1198651into

1198652 and 120587

1into 120587

2 119891lowast1205792= 1205791and (119889119891) ∘ 120593

1= 1205932∘ (119889119891)

We now state and prove themain result of this paperThisresult gives equivalent assertions between the condition (C3)twist 120579 of (119889120578 119868) and contact maps for a contact groupoidover119872

Theorem 16 Let 119872 be a generalized almost contact manifoldand (Σ 120578 119868) an induced contact groupoid over 119872 with theinduced multiplicative (1 1)-tensor Assume that ((119865 120587) 119868)

satisfy (C1) (C2) with integrable (119865 120587) Then for a 120579 2-formon119872 the following assertions are equivalent

(i) (C3) is satisfied(ii) 119889120578 + 119868

lowast119889120578 minus 120578 and (119894119865119889120578) = 119904lowast120579 minus 119905lowast120579

(iii) (119905 119904) Σ rarr 119872 times 119872 is a generalized contact mapcondition of generalized contactmap on119872 is (119889119905)∘120593

1=

1205932∘ (119889119905) this condition on119872 is (119889119904) ∘ 120593

1= minus1205932∘ (119889119904)

Proof (i) hArr (ii) define 120601 = 120579minus119905lowast120579+119904lowast120579 such that 120579 = minus119889120578minus

119868lowast119889120578+120578and(119894119865119889120578) is the twist of (119889120578 119868) and119860 = ker(119889119904)|

119872 We

know fromTheorem 1 that closed multiplicative 2-form 120595 onΣ vanishes if and only if 119868119872 form 119906

120595= 0 that is120595(119883 120572) = 0

such that119883 isin 119879119872 120572 isin 119860 This case can be applied for formswith higher degree that is 3-form 120595 vanishes if and only if120595(119883 119884 120572) = 0

Since 119889120578 and (119889120578)119868

are closed from (35) we get119894119883and119884

(119889(119868lowast119889120578)) = minus119894119873119868(119883119884)

119889120578 Putting 120579 = minus119889120578 minus 119868lowast119889120578 + 120578 and

(119894119865119889120578) we obtain

119894119883and119884

119889 (minus120579 + 120578 and (119894119865119889120578)) = minus119894

119873119868(119883119884)119889120578 (51)

Since 119889120601 = 0 hArr 119889120601(119883 119884 120572) = 0 we have

119889120601 (119883 119884 120572) = 0 lArrrArr 119889120579 (119883 119884 120572) minus 119889 (119905lowast120579) (119883 119884 120572)

+ 119889 (119904lowast120579) (119883 119884 120572) = 0

(52)

On the other hand we obtain

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119889119905 (119883) 119889119905 (119884) 119889119905 (120572)) (53)

If we take 119889119905 = 120588 in (53) for 119860 we get

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119889119905 (119883) 119889119905 (119884) 120588 (120572)) (54)

On the other hand from [17] we know that

119868119889Σ= 119898 ∘ (119905 119868119889

Σ) (55)

Differentiating (55) we obtain

119883 = 119889119905 (119883) (56)

Using (56) in (54) we get

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119883 119884 120588 (120572)) (57)

In a similar way we see that

119889 (119904lowast120579) (119883 119884 120572) = 119889120579 (119889119904 (119883) 119889119904 (119884) 119889119904 (120572)) (58)

Since 120572 isin ker 119889119904 then 119889119904(120572) = 0 Hence 119889(119904lowast120579) = 0 Thuswe obtain

119889120579 (119883 119884 120572) = 119889120579 (119883 119884 120588 (120572)) (59)

Using (51) in (59) we derive

119889 (120578 and (119894119865119889120578)) (119883 119884 120572) + 119889120578 (119873

119868(119883 119884) 120572)

= 119889120579 (119883 119884 120588 (120572)) (60)

Abstract and Applied Analysis 7

On the other hand it is clear that 120601 = 0 hArr 120579 minus 119905lowast120579 + 119904lowast120579 = 0Thus we obtain

120579 (119883 120572) = 120579 (119883 120588 (120572)) (61)

Since 120579 = minus119889120578 minus 119868lowast119889120578 + 120578 and (119894119865119889120578) we get

minus 119889120578 (119883 120572) minus 119889120578 (119868119883 119868120572) + (120578 and (119894119865119889120578)) (119883 120572)

= 120579 (119883 120588 (120572)) (62)

Since Jacobi structure (119865 120587) is integrable it defines a Liealgebroid whose anchor map is 120588 = (120587 119865)

♯ Let us use (120587 119865)♯instead of 120588 in (60) and (62) then we get

119889 (120578 and (119894119865119889120578)) (119883 119884 120572) + 119889120578 (119873

119868(119883 119884) 120572)

= 119889120579 (119883 119884 120587♯(120572) + 119891119865)

(63)

minus 119889120578 (119883 120572) minus 119889120578 (119868119883 119868120572) + (120578 and (119894119865119889120578)) (119883 120572)

= 120579 (119883 120587♯(120572) + 119891119865)

(64)

Since 119889120578(120572119883) = 120572(119883) (119889120578)119868(120572 119883) = 120572(120593119883) from (63) we

have

minus 120572 (119889120578 (119883 119884) 119865) minus 120572 (119873120593(119883 119884))

= 119889120579 (119883 119884 120587♯(120572) + 119891119865)

= 119894119883and119884

119889120579 (120587♯(120572) + 119891119865)

= 119894119883and119884

119889120579 (120587♯(120572)) + 119894

119883and119884119889120579 (119891119865)

= 120587 (120572 119894119883and119884

119889120579)

= minus120572 (120587♯(119894119883and119884

119889120579))

(65)

that is 120572(119889120578(119883 119884)119865 + 119873120593(119883 119884)) = 120572(120587♯(119894

119883and119884119889120579))

Since the above equation holds for all nondegenerate 120572we get

119889120578 (119883 119884) 119865 + 119873120593 (119883 119884) = 120587

♯(119894119883and119884

119889120579) (66)

Then we arrive at

119889120578 (120593119883 120593119884) 119865 + 119873120593 (119883 119884) = 120587

♯(119894119883and119884

119889120579) (67)

On the other hand from (64) we obtain

120572 (119883) + 120572 (1205932119883) minus 120578 (119883) 120572 (119865) = 120587 (120572 119894

119883120579) + 119894119883120579 (119891119865)

= minus120572 (120587♯120579♯119883)

(68)

Thus we get

1205932+ 120587♯120579♯= minus119868119889 + 119865 ⊙ 120578 (69)

Then (i)hArr(ii) follows from (67) and (69)

(ii) hArr (iii) 119889120578+119868lowast119889120578minus120578and119894

119865119889120578 = 119904lowast120579minus119905lowast120579 says that (119905 119904)

is compatible with 2-form 119889120578 Also it is clear that (119905 119904) andbivectors are compatible because Σ is a contact groupoid Wewill check the compatibility of (119905 119904) and (1 1)-tensors Fromcompatibility condition of 119905 and 119904 we get 119889119905 ∘ 119868 = 120593 ∘ 119889119905 and119889119904 ∘ 119868 = minus120593 ∘ 119889119904

For all 120572 isin 119860 119881 isin 120594(Σ) and 119891 isin R we have

119889120578 (120572 119881) = 119889120578 (120572 119889119905119881) (70)

which is equivalent to

120572 (119881) = ⟨119906(119889120578)

(120572) 119889119905119881⟩ (71)

Since 119906(119889120578)

= 119868119889 and 119906((119889120578)119868)

= 120593lowast ∘ 1199011199031such that 119906

((119889120578)119868)(120572) =

120593lowast ∘ 1199011199031(120572 119891) = 120593lowast(120572) we get

⟨120572 120593 (119889119905 (119881))⟩ = 120572 (120593 (119889119905 (119881)))

= 120593lowast120572 (119889119905 (119881))

= ⟨119906((119889120578)119868)

120572 119881⟩

= 119889120578 (120572 119868119881)

= ⟨120572 119889119905 (119868119881)⟩

(72)

Since this equation holds for all 120572 isin 119860 119886(119889119905) = 119889119905(119868) Using119904 = 119905 ∘ 119894

119886 (119889119904 (119881)) = 119886119889 (119905 ∘ 119894) 119881

= 120593 (119889119905 (119889119894 (119881)))

= minus120593 (119889119905 (119881))

= minus119889119904 (119868119881)

(73)

which shows that 119886(119889119904) = minus119889119904(119868) Thus proof is completed

Conflict of Interests

The author declares that she has no conflict of interests

References

[1] C Ehresmann ldquoCategories topologiques et categoriesdifferentiablesrdquo in Colloque de Geometrie Differentielle GlobaleBruxelles Belgium pp 137ndash150 Librairie Universitaire 1959

[2] M Crainic and R L Fernandes ldquoIntegrability of Lie bracketsrdquoAnnals of Mathematics Second Series vol 157 no 2 pp 575ndash620 2003

[3] M Degeratu ldquoProjective limits of Lie groupoidsrdquo AnaleleUniversitatii din Oradea Fascicola Matematica vol 16 pp 183ndash188 2009

[4] M R Farhangdoost and T Nasirzade ldquoGeometrical categoriesof generalized Lie groups and Lie group-groupoidsrdquo IranianJournal of Science and Technology A Science vol 37 no 1 pp69ndash73 2013

[5] C-M Marle ldquoCalculus on Lie algebroids Lie groupoids andPoisson manifoldsrdquo Dissertationes Mathematicae (RozprawyMatematyczne) vol 457 p 57 2008

8 Abstract and Applied Analysis

[6] D S Zhong Z Chen and Z J Liu ldquoOn the existence of globalbisections of Lie groupoidsrdquo Acta Mathematica Sinica (EnglishSeries) vol 25 no 6 pp 1001ndash1014 2009

[7] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiablesCalcul differenetiel dans la categorie des groupoıdesinfinitesimauxrdquo Comptes Rendus de lrsquoAcademie des Sciences Avol 264 pp 245ndash248 1967

[8] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquoThe QuarterlyJournal of Mathematics vol 54 no 3 pp 281ndash308 2003

[9] M Gualtieri Generalized complex geometry [PhD thesis]University of Oxford 2003

[10] D Iglesias-Ponte and A Wade ldquoContact manifolds and gener-alized complex structuresrdquo Journal of Geometry and Physics vol53 no 3 pp 249ndash258 2005

[11] Y S Poon and A Wade ldquoGeneralized contact structuresrdquoJournal of the London Mathematical Society Second Series vol83 no 2 pp 333ndash352 2011

[12] A Wade ldquoLocal structure of generalized contact manifoldsrdquoDifferential Geometry and its Applications vol 30 no 1 pp 124ndash135 2012

[13] M Crainic ldquoGeneralized complex structures and Lie bracketsrdquoBulletin of the Brazilian Mathematical Society New SeriesBoletim da Sociedade Brasileira de Matematica vol 42 no 4pp 559ndash578 2011

[14] M Crainic and C Zhu ldquoIntegrability of Jacobi structuresrdquoAnnales de lrsquoinstitut Fourier vol 57 no 4 pp 1181ndash1216 2007

[15] A A Kirillov ldquoLocal Lie algebrasrdquo Russian MathematicalSurveys vol 31 pp 55ndash75 1976

[16] M Crainic and R L Fernandes ldquoLectures on integrability of Liebracketsrdquo in Lectures on Poisson Geometry vol 17 of Geometryand Topology Monographs pp 1ndash107 Geometry amp TopologyPublications Coventry UK 2011

[17] H BursztynM Crainic AWeinstein and C Zhu ldquoIntegrationof twisted Dirac bracketsrdquo Duke Mathematical Journal vol 123no 3 pp 549ndash607 2004

[18] J Janyska and M Modugno ldquoGeneralized geometrical struc-tures of odd dimensional manifoldsrdquo Journal de MathematiquesPures et Appliquees Neuvieme Serie vol 91 no 2 pp 211ndash2322009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Using (35) then we get

119894119883and119884

(119889 ((120593lowast(119889120578))120593))

= 119894120593119883and119884+119883and120593119884

(119889 (120593lowast119889120578)) minus 119894

119873120593(119883119884)(119889120578)120593

(48)

Since 119894119873(119883119884)

((119889120578)120593) = 120593lowast119894

119873(119883119884)(119889120578) applying 119894

119883and119884119889(120593lowast119889120578)

= minus119894119873120593(119883119884)

(119889120578) to (48) we have

119894119883and119884

(119889 ((120593lowast(119889120578))120593))

= 119894120593119883and119884+119883and120593119884

(119889 (120593lowast119889120578)) + 120593

lowast(119894119883and119884

(119889 (120593lowast119889120578)))

(49)

In a similar way we can obtain (C5)The converse is clear from Lemmas 10 and 13

We note that similar to 2-forms given a Lie groupoid 119866a (1 1)-tensor 119869 119879119866 rarr 119879119866 is called multiplicative [13] iffor any (119892 ℎ) isin 119866 times 119866 and any V

119892isin 119879119892119866 119908ℎisin 119879ℎ119866 such that

(V119892 119908ℎ) is tangent to 119866 times 119866 at (119892 ℎ) so is (119869V

119892 119869119908ℎ) and

(119889119898)119892ℎ

(119869V119892 119869119908ℎ) = 119869 ((119889119898)

119892ℎ(V119892 119908ℎ)) (50)

Let (119872 120578) be a contact manifold Then it is easy to see thatthere is a one to one correspondence between (1 1)-tensors120593 commuting with 119889120578 and 2-forms on119872 On the other handit is easy to see that (C2) is equivalent to the fact that 120593lowast ∘ 119901119903

1

is an 119868119872 form on the Lie algebroid119879lowast119872oplusR associated Jacobistructure (119865 120587) Thus from the above discussion Lemma 13andTheorem 1 one can concludewith the following theorem

Theorem 15 Let119872 be a generalized almost contact manifoldLet (119865 120587) be an integrable Jacobi structure on 119872 and (Σ 120578)

a contact groupoid over 119872 Then there is a natural 1-1correspondence between

(i) (1 1)-tensors 120593 on119872 satisfying (C2)

(ii) multiplicative (1 1)-tensors 119868 on Σ with the propertythat (119868 119889120578) is a Hitchin pair

We recall the notion of generalized contact map betweengeneralized contact manifolds This notion is similar to thegeneralized holomorphic map given in [13]

Let (119872119894I119894) 119894 = 1 2 be two generalized contact

manifolds and let 120593119894 120587119894 120579119894be the components of I

119894in the

matrix representation (11) A map 119891 1198721

rarr 1198722is called

generalized contact if and only if 119891 maps 1205931into 120593

2 1198651into

1198652 and 120587

1into 120587

2 119891lowast1205792= 1205791and (119889119891) ∘ 120593

1= 1205932∘ (119889119891)

We now state and prove themain result of this paperThisresult gives equivalent assertions between the condition (C3)twist 120579 of (119889120578 119868) and contact maps for a contact groupoidover119872

Theorem 16 Let 119872 be a generalized almost contact manifoldand (Σ 120578 119868) an induced contact groupoid over 119872 with theinduced multiplicative (1 1)-tensor Assume that ((119865 120587) 119868)

satisfy (C1) (C2) with integrable (119865 120587) Then for a 120579 2-formon119872 the following assertions are equivalent

(i) (C3) is satisfied(ii) 119889120578 + 119868

lowast119889120578 minus 120578 and (119894119865119889120578) = 119904lowast120579 minus 119905lowast120579

(iii) (119905 119904) Σ rarr 119872 times 119872 is a generalized contact mapcondition of generalized contactmap on119872 is (119889119905)∘120593

1=

1205932∘ (119889119905) this condition on119872 is (119889119904) ∘ 120593

1= minus1205932∘ (119889119904)

Proof (i) hArr (ii) define 120601 = 120579minus119905lowast120579+119904lowast120579 such that 120579 = minus119889120578minus

119868lowast119889120578+120578and(119894119865119889120578) is the twist of (119889120578 119868) and119860 = ker(119889119904)|

119872 We

know fromTheorem 1 that closed multiplicative 2-form 120595 onΣ vanishes if and only if 119868119872 form 119906

120595= 0 that is120595(119883 120572) = 0

such that119883 isin 119879119872 120572 isin 119860 This case can be applied for formswith higher degree that is 3-form 120595 vanishes if and only if120595(119883 119884 120572) = 0

Since 119889120578 and (119889120578)119868

are closed from (35) we get119894119883and119884

(119889(119868lowast119889120578)) = minus119894119873119868(119883119884)

119889120578 Putting 120579 = minus119889120578 minus 119868lowast119889120578 + 120578 and

(119894119865119889120578) we obtain

119894119883and119884

119889 (minus120579 + 120578 and (119894119865119889120578)) = minus119894

119873119868(119883119884)119889120578 (51)

Since 119889120601 = 0 hArr 119889120601(119883 119884 120572) = 0 we have

119889120601 (119883 119884 120572) = 0 lArrrArr 119889120579 (119883 119884 120572) minus 119889 (119905lowast120579) (119883 119884 120572)

+ 119889 (119904lowast120579) (119883 119884 120572) = 0

(52)

On the other hand we obtain

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119889119905 (119883) 119889119905 (119884) 119889119905 (120572)) (53)

If we take 119889119905 = 120588 in (53) for 119860 we get

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119889119905 (119883) 119889119905 (119884) 120588 (120572)) (54)

On the other hand from [17] we know that

119868119889Σ= 119898 ∘ (119905 119868119889

Σ) (55)

Differentiating (55) we obtain

119883 = 119889119905 (119883) (56)

Using (56) in (54) we get

119889 (119905lowast120579) (119883 119884 120572) = 119889120579 (119883 119884 120588 (120572)) (57)

In a similar way we see that

119889 (119904lowast120579) (119883 119884 120572) = 119889120579 (119889119904 (119883) 119889119904 (119884) 119889119904 (120572)) (58)

Since 120572 isin ker 119889119904 then 119889119904(120572) = 0 Hence 119889(119904lowast120579) = 0 Thuswe obtain

119889120579 (119883 119884 120572) = 119889120579 (119883 119884 120588 (120572)) (59)

Using (51) in (59) we derive

119889 (120578 and (119894119865119889120578)) (119883 119884 120572) + 119889120578 (119873

119868(119883 119884) 120572)

= 119889120579 (119883 119884 120588 (120572)) (60)

Abstract and Applied Analysis 7

On the other hand it is clear that 120601 = 0 hArr 120579 minus 119905lowast120579 + 119904lowast120579 = 0Thus we obtain

120579 (119883 120572) = 120579 (119883 120588 (120572)) (61)

Since 120579 = minus119889120578 minus 119868lowast119889120578 + 120578 and (119894119865119889120578) we get

minus 119889120578 (119883 120572) minus 119889120578 (119868119883 119868120572) + (120578 and (119894119865119889120578)) (119883 120572)

= 120579 (119883 120588 (120572)) (62)

Since Jacobi structure (119865 120587) is integrable it defines a Liealgebroid whose anchor map is 120588 = (120587 119865)

♯ Let us use (120587 119865)♯instead of 120588 in (60) and (62) then we get

119889 (120578 and (119894119865119889120578)) (119883 119884 120572) + 119889120578 (119873

119868(119883 119884) 120572)

= 119889120579 (119883 119884 120587♯(120572) + 119891119865)

(63)

minus 119889120578 (119883 120572) minus 119889120578 (119868119883 119868120572) + (120578 and (119894119865119889120578)) (119883 120572)

= 120579 (119883 120587♯(120572) + 119891119865)

(64)

Since 119889120578(120572119883) = 120572(119883) (119889120578)119868(120572 119883) = 120572(120593119883) from (63) we

have

minus 120572 (119889120578 (119883 119884) 119865) minus 120572 (119873120593(119883 119884))

= 119889120579 (119883 119884 120587♯(120572) + 119891119865)

= 119894119883and119884

119889120579 (120587♯(120572) + 119891119865)

= 119894119883and119884

119889120579 (120587♯(120572)) + 119894

119883and119884119889120579 (119891119865)

= 120587 (120572 119894119883and119884

119889120579)

= minus120572 (120587♯(119894119883and119884

119889120579))

(65)

that is 120572(119889120578(119883 119884)119865 + 119873120593(119883 119884)) = 120572(120587♯(119894

119883and119884119889120579))

Since the above equation holds for all nondegenerate 120572we get

119889120578 (119883 119884) 119865 + 119873120593 (119883 119884) = 120587

♯(119894119883and119884

119889120579) (66)

Then we arrive at

119889120578 (120593119883 120593119884) 119865 + 119873120593 (119883 119884) = 120587

♯(119894119883and119884

119889120579) (67)

On the other hand from (64) we obtain

120572 (119883) + 120572 (1205932119883) minus 120578 (119883) 120572 (119865) = 120587 (120572 119894

119883120579) + 119894119883120579 (119891119865)

= minus120572 (120587♯120579♯119883)

(68)

Thus we get

1205932+ 120587♯120579♯= minus119868119889 + 119865 ⊙ 120578 (69)

Then (i)hArr(ii) follows from (67) and (69)

(ii) hArr (iii) 119889120578+119868lowast119889120578minus120578and119894

119865119889120578 = 119904lowast120579minus119905lowast120579 says that (119905 119904)

is compatible with 2-form 119889120578 Also it is clear that (119905 119904) andbivectors are compatible because Σ is a contact groupoid Wewill check the compatibility of (119905 119904) and (1 1)-tensors Fromcompatibility condition of 119905 and 119904 we get 119889119905 ∘ 119868 = 120593 ∘ 119889119905 and119889119904 ∘ 119868 = minus120593 ∘ 119889119904

For all 120572 isin 119860 119881 isin 120594(Σ) and 119891 isin R we have

119889120578 (120572 119881) = 119889120578 (120572 119889119905119881) (70)

which is equivalent to

120572 (119881) = ⟨119906(119889120578)

(120572) 119889119905119881⟩ (71)

Since 119906(119889120578)

= 119868119889 and 119906((119889120578)119868)

= 120593lowast ∘ 1199011199031such that 119906

((119889120578)119868)(120572) =

120593lowast ∘ 1199011199031(120572 119891) = 120593lowast(120572) we get

⟨120572 120593 (119889119905 (119881))⟩ = 120572 (120593 (119889119905 (119881)))

= 120593lowast120572 (119889119905 (119881))

= ⟨119906((119889120578)119868)

120572 119881⟩

= 119889120578 (120572 119868119881)

= ⟨120572 119889119905 (119868119881)⟩

(72)

Since this equation holds for all 120572 isin 119860 119886(119889119905) = 119889119905(119868) Using119904 = 119905 ∘ 119894

119886 (119889119904 (119881)) = 119886119889 (119905 ∘ 119894) 119881

= 120593 (119889119905 (119889119894 (119881)))

= minus120593 (119889119905 (119881))

= minus119889119904 (119868119881)

(73)

which shows that 119886(119889119904) = minus119889119904(119868) Thus proof is completed

Conflict of Interests

The author declares that she has no conflict of interests

References

[1] C Ehresmann ldquoCategories topologiques et categoriesdifferentiablesrdquo in Colloque de Geometrie Differentielle GlobaleBruxelles Belgium pp 137ndash150 Librairie Universitaire 1959

[2] M Crainic and R L Fernandes ldquoIntegrability of Lie bracketsrdquoAnnals of Mathematics Second Series vol 157 no 2 pp 575ndash620 2003

[3] M Degeratu ldquoProjective limits of Lie groupoidsrdquo AnaleleUniversitatii din Oradea Fascicola Matematica vol 16 pp 183ndash188 2009

[4] M R Farhangdoost and T Nasirzade ldquoGeometrical categoriesof generalized Lie groups and Lie group-groupoidsrdquo IranianJournal of Science and Technology A Science vol 37 no 1 pp69ndash73 2013

[5] C-M Marle ldquoCalculus on Lie algebroids Lie groupoids andPoisson manifoldsrdquo Dissertationes Mathematicae (RozprawyMatematyczne) vol 457 p 57 2008

8 Abstract and Applied Analysis

[6] D S Zhong Z Chen and Z J Liu ldquoOn the existence of globalbisections of Lie groupoidsrdquo Acta Mathematica Sinica (EnglishSeries) vol 25 no 6 pp 1001ndash1014 2009

[7] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiablesCalcul differenetiel dans la categorie des groupoıdesinfinitesimauxrdquo Comptes Rendus de lrsquoAcademie des Sciences Avol 264 pp 245ndash248 1967

[8] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquoThe QuarterlyJournal of Mathematics vol 54 no 3 pp 281ndash308 2003

[9] M Gualtieri Generalized complex geometry [PhD thesis]University of Oxford 2003

[10] D Iglesias-Ponte and A Wade ldquoContact manifolds and gener-alized complex structuresrdquo Journal of Geometry and Physics vol53 no 3 pp 249ndash258 2005

[11] Y S Poon and A Wade ldquoGeneralized contact structuresrdquoJournal of the London Mathematical Society Second Series vol83 no 2 pp 333ndash352 2011

[12] A Wade ldquoLocal structure of generalized contact manifoldsrdquoDifferential Geometry and its Applications vol 30 no 1 pp 124ndash135 2012

[13] M Crainic ldquoGeneralized complex structures and Lie bracketsrdquoBulletin of the Brazilian Mathematical Society New SeriesBoletim da Sociedade Brasileira de Matematica vol 42 no 4pp 559ndash578 2011

[14] M Crainic and C Zhu ldquoIntegrability of Jacobi structuresrdquoAnnales de lrsquoinstitut Fourier vol 57 no 4 pp 1181ndash1216 2007

[15] A A Kirillov ldquoLocal Lie algebrasrdquo Russian MathematicalSurveys vol 31 pp 55ndash75 1976

[16] M Crainic and R L Fernandes ldquoLectures on integrability of Liebracketsrdquo in Lectures on Poisson Geometry vol 17 of Geometryand Topology Monographs pp 1ndash107 Geometry amp TopologyPublications Coventry UK 2011

[17] H BursztynM Crainic AWeinstein and C Zhu ldquoIntegrationof twisted Dirac bracketsrdquo Duke Mathematical Journal vol 123no 3 pp 549ndash607 2004

[18] J Janyska and M Modugno ldquoGeneralized geometrical struc-tures of odd dimensional manifoldsrdquo Journal de MathematiquesPures et Appliquees Neuvieme Serie vol 91 no 2 pp 211ndash2322009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

On the other hand it is clear that 120601 = 0 hArr 120579 minus 119905lowast120579 + 119904lowast120579 = 0Thus we obtain

120579 (119883 120572) = 120579 (119883 120588 (120572)) (61)

Since 120579 = minus119889120578 minus 119868lowast119889120578 + 120578 and (119894119865119889120578) we get

minus 119889120578 (119883 120572) minus 119889120578 (119868119883 119868120572) + (120578 and (119894119865119889120578)) (119883 120572)

= 120579 (119883 120588 (120572)) (62)

Since Jacobi structure (119865 120587) is integrable it defines a Liealgebroid whose anchor map is 120588 = (120587 119865)

♯ Let us use (120587 119865)♯instead of 120588 in (60) and (62) then we get

119889 (120578 and (119894119865119889120578)) (119883 119884 120572) + 119889120578 (119873

119868(119883 119884) 120572)

= 119889120579 (119883 119884 120587♯(120572) + 119891119865)

(63)

minus 119889120578 (119883 120572) minus 119889120578 (119868119883 119868120572) + (120578 and (119894119865119889120578)) (119883 120572)

= 120579 (119883 120587♯(120572) + 119891119865)

(64)

Since 119889120578(120572119883) = 120572(119883) (119889120578)119868(120572 119883) = 120572(120593119883) from (63) we

have

minus 120572 (119889120578 (119883 119884) 119865) minus 120572 (119873120593(119883 119884))

= 119889120579 (119883 119884 120587♯(120572) + 119891119865)

= 119894119883and119884

119889120579 (120587♯(120572) + 119891119865)

= 119894119883and119884

119889120579 (120587♯(120572)) + 119894

119883and119884119889120579 (119891119865)

= 120587 (120572 119894119883and119884

119889120579)

= minus120572 (120587♯(119894119883and119884

119889120579))

(65)

that is 120572(119889120578(119883 119884)119865 + 119873120593(119883 119884)) = 120572(120587♯(119894

119883and119884119889120579))

Since the above equation holds for all nondegenerate 120572we get

119889120578 (119883 119884) 119865 + 119873120593 (119883 119884) = 120587

♯(119894119883and119884

119889120579) (66)

Then we arrive at

119889120578 (120593119883 120593119884) 119865 + 119873120593 (119883 119884) = 120587

♯(119894119883and119884

119889120579) (67)

On the other hand from (64) we obtain

120572 (119883) + 120572 (1205932119883) minus 120578 (119883) 120572 (119865) = 120587 (120572 119894

119883120579) + 119894119883120579 (119891119865)

= minus120572 (120587♯120579♯119883)

(68)

Thus we get

1205932+ 120587♯120579♯= minus119868119889 + 119865 ⊙ 120578 (69)

Then (i)hArr(ii) follows from (67) and (69)

(ii) hArr (iii) 119889120578+119868lowast119889120578minus120578and119894

119865119889120578 = 119904lowast120579minus119905lowast120579 says that (119905 119904)

is compatible with 2-form 119889120578 Also it is clear that (119905 119904) andbivectors are compatible because Σ is a contact groupoid Wewill check the compatibility of (119905 119904) and (1 1)-tensors Fromcompatibility condition of 119905 and 119904 we get 119889119905 ∘ 119868 = 120593 ∘ 119889119905 and119889119904 ∘ 119868 = minus120593 ∘ 119889119904

For all 120572 isin 119860 119881 isin 120594(Σ) and 119891 isin R we have

119889120578 (120572 119881) = 119889120578 (120572 119889119905119881) (70)

which is equivalent to

120572 (119881) = ⟨119906(119889120578)

(120572) 119889119905119881⟩ (71)

Since 119906(119889120578)

= 119868119889 and 119906((119889120578)119868)

= 120593lowast ∘ 1199011199031such that 119906

((119889120578)119868)(120572) =

120593lowast ∘ 1199011199031(120572 119891) = 120593lowast(120572) we get

⟨120572 120593 (119889119905 (119881))⟩ = 120572 (120593 (119889119905 (119881)))

= 120593lowast120572 (119889119905 (119881))

= ⟨119906((119889120578)119868)

120572 119881⟩

= 119889120578 (120572 119868119881)

= ⟨120572 119889119905 (119868119881)⟩

(72)

Since this equation holds for all 120572 isin 119860 119886(119889119905) = 119889119905(119868) Using119904 = 119905 ∘ 119894

119886 (119889119904 (119881)) = 119886119889 (119905 ∘ 119894) 119881

= 120593 (119889119905 (119889119894 (119881)))

= minus120593 (119889119905 (119881))

= minus119889119904 (119868119881)

(73)

which shows that 119886(119889119904) = minus119889119904(119868) Thus proof is completed

Conflict of Interests

The author declares that she has no conflict of interests

References

[1] C Ehresmann ldquoCategories topologiques et categoriesdifferentiablesrdquo in Colloque de Geometrie Differentielle GlobaleBruxelles Belgium pp 137ndash150 Librairie Universitaire 1959

[2] M Crainic and R L Fernandes ldquoIntegrability of Lie bracketsrdquoAnnals of Mathematics Second Series vol 157 no 2 pp 575ndash620 2003

[3] M Degeratu ldquoProjective limits of Lie groupoidsrdquo AnaleleUniversitatii din Oradea Fascicola Matematica vol 16 pp 183ndash188 2009

[4] M R Farhangdoost and T Nasirzade ldquoGeometrical categoriesof generalized Lie groups and Lie group-groupoidsrdquo IranianJournal of Science and Technology A Science vol 37 no 1 pp69ndash73 2013

[5] C-M Marle ldquoCalculus on Lie algebroids Lie groupoids andPoisson manifoldsrdquo Dissertationes Mathematicae (RozprawyMatematyczne) vol 457 p 57 2008

8 Abstract and Applied Analysis

[6] D S Zhong Z Chen and Z J Liu ldquoOn the existence of globalbisections of Lie groupoidsrdquo Acta Mathematica Sinica (EnglishSeries) vol 25 no 6 pp 1001ndash1014 2009

[7] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiablesCalcul differenetiel dans la categorie des groupoıdesinfinitesimauxrdquo Comptes Rendus de lrsquoAcademie des Sciences Avol 264 pp 245ndash248 1967

[8] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquoThe QuarterlyJournal of Mathematics vol 54 no 3 pp 281ndash308 2003

[9] M Gualtieri Generalized complex geometry [PhD thesis]University of Oxford 2003

[10] D Iglesias-Ponte and A Wade ldquoContact manifolds and gener-alized complex structuresrdquo Journal of Geometry and Physics vol53 no 3 pp 249ndash258 2005

[11] Y S Poon and A Wade ldquoGeneralized contact structuresrdquoJournal of the London Mathematical Society Second Series vol83 no 2 pp 333ndash352 2011

[12] A Wade ldquoLocal structure of generalized contact manifoldsrdquoDifferential Geometry and its Applications vol 30 no 1 pp 124ndash135 2012

[13] M Crainic ldquoGeneralized complex structures and Lie bracketsrdquoBulletin of the Brazilian Mathematical Society New SeriesBoletim da Sociedade Brasileira de Matematica vol 42 no 4pp 559ndash578 2011

[14] M Crainic and C Zhu ldquoIntegrability of Jacobi structuresrdquoAnnales de lrsquoinstitut Fourier vol 57 no 4 pp 1181ndash1216 2007

[15] A A Kirillov ldquoLocal Lie algebrasrdquo Russian MathematicalSurveys vol 31 pp 55ndash75 1976

[16] M Crainic and R L Fernandes ldquoLectures on integrability of Liebracketsrdquo in Lectures on Poisson Geometry vol 17 of Geometryand Topology Monographs pp 1ndash107 Geometry amp TopologyPublications Coventry UK 2011

[17] H BursztynM Crainic AWeinstein and C Zhu ldquoIntegrationof twisted Dirac bracketsrdquo Duke Mathematical Journal vol 123no 3 pp 549ndash607 2004

[18] J Janyska and M Modugno ldquoGeneralized geometrical struc-tures of odd dimensional manifoldsrdquo Journal de MathematiquesPures et Appliquees Neuvieme Serie vol 91 no 2 pp 211ndash2322009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

[6] D S Zhong Z Chen and Z J Liu ldquoOn the existence of globalbisections of Lie groupoidsrdquo Acta Mathematica Sinica (EnglishSeries) vol 25 no 6 pp 1001ndash1014 2009

[7] J Pradines ldquoTheorie de Lie pour les groupoıdes differentiablesCalcul differenetiel dans la categorie des groupoıdesinfinitesimauxrdquo Comptes Rendus de lrsquoAcademie des Sciences Avol 264 pp 245ndash248 1967

[8] N Hitchin ldquoGeneralized Calabi-Yau manifoldsrdquoThe QuarterlyJournal of Mathematics vol 54 no 3 pp 281ndash308 2003

[9] M Gualtieri Generalized complex geometry [PhD thesis]University of Oxford 2003

[10] D Iglesias-Ponte and A Wade ldquoContact manifolds and gener-alized complex structuresrdquo Journal of Geometry and Physics vol53 no 3 pp 249ndash258 2005

[11] Y S Poon and A Wade ldquoGeneralized contact structuresrdquoJournal of the London Mathematical Society Second Series vol83 no 2 pp 333ndash352 2011

[12] A Wade ldquoLocal structure of generalized contact manifoldsrdquoDifferential Geometry and its Applications vol 30 no 1 pp 124ndash135 2012

[13] M Crainic ldquoGeneralized complex structures and Lie bracketsrdquoBulletin of the Brazilian Mathematical Society New SeriesBoletim da Sociedade Brasileira de Matematica vol 42 no 4pp 559ndash578 2011

[14] M Crainic and C Zhu ldquoIntegrability of Jacobi structuresrdquoAnnales de lrsquoinstitut Fourier vol 57 no 4 pp 1181ndash1216 2007

[15] A A Kirillov ldquoLocal Lie algebrasrdquo Russian MathematicalSurveys vol 31 pp 55ndash75 1976

[16] M Crainic and R L Fernandes ldquoLectures on integrability of Liebracketsrdquo in Lectures on Poisson Geometry vol 17 of Geometryand Topology Monographs pp 1ndash107 Geometry amp TopologyPublications Coventry UK 2011

[17] H BursztynM Crainic AWeinstein and C Zhu ldquoIntegrationof twisted Dirac bracketsrdquo Duke Mathematical Journal vol 123no 3 pp 549ndash607 2004

[18] J Janyska and M Modugno ldquoGeneralized geometrical struc-tures of odd dimensional manifoldsrdquo Journal de MathematiquesPures et Appliquees Neuvieme Serie vol 91 no 2 pp 211ndash2322009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of