Left invariant Randers metrics on 3-dimensionalHeisenberg group

Z. Kovacs1,2 and A. Lengyelne-Toth1

1University of Debrecen

2College of Nyıregyhaza

Symposium on Finsler Geometry, 2013 Sapporo

Summary

The purpose of this talk is to generalise some basic results ongeometry of 2-step nilpotent groups with left invariant Riemannianmetric to left invariant Randers metric. (Proposed by P. T. Nagy.)

(N , 〈, 〉) 2-step nilpotent Lie algebra with positive definiteinner product ([[N ,N ],N ] = 0).

(N, 〈, 〉) the simply connected 2-step nilpotent group N withLie algebra N and left invariant metric 〈, 〉: the lefttranslations La, a ∈ N are isometries.

P. Eberlein.

Geometry of 2-step nilpotent groups with a left invariant metric.

Ann. Sci. Ec. Norm. Super., IV. Ser., 27(5):611–660, 1994.

Why study left invariant metrics?

The geometry of any Lie group G with left invariant Riemannianmetric reflects strongly the algebraic structure of the Lie algebra G.

Theorem (Milnor)

If Z belongs to the center of the Lie algebra G, then for any leftinvariant metric the inequality K (Z ,X ) ≥ 0 is satisfied for all X .

Remark

Sectional curvatures associated with a bi-invariant metric can becomputed by the explicit formula K (U,V ) = 1

4 〈[U,V ], [U,V ]〉.

J. Milnor.

Curvatures of left invariant metrics on Lie groups.

Advances in Math., 21:293–329, 1976.

2-step nilpotent groups with left invariant RiemannianmetricSectional curvature

Theorem (Wolf, 1964)

Any nonabelian nilpotent Lie group with left invariant metric mustadmit both positive and negative sectional curvatures. (Milnorextended the result to Ricci curvature.)

Theorem (Eberlein, 1994) for 2-step nilpotent groups

Π = span(X ,Y ) ⊆ TnN, where (X ,Y ) is orthonormal pair.K (Π) = K (X ,Y ) = 〈R(X ,Y )Y ,X 〉.

K (X ,Y ) = −3

4‖[X ,Y ]‖2, X ,Y ∈ V

K (X ,Z ) =1

4‖j(Z )X‖2, X ∈ V,Z ∈ Z

K (Z ,Z ∗) = 0, Z ,Z ∗ ∈ Z

2-step nilpotent groups with left invariant RiemannianmetricLevi-Civita connection

Let (N = V ⊕ Z, 〈, 〉) denote a 2-step nilpotent Lie algebra withpositive definite inner product. Let (N, 〈, 〉) denote the simplyconnected 2-step nilpotent group N with Lie algebra N and leftinvariant metric 〈, 〉. The Levi-Civita connection ∇ is a bilinearmap from N ×N into N .

∇XY =1

2[X ,Y ] (X ,Y ∈ V)

∇XZ = ∇ZX = −1

2j(Z )X (X ∈ V,Z ∈ Z)

∇ZZ ∗ = 0 (Z ,Z ∗ ∈ Z),

where ∀Z ∈ Z, j(Z ) : V → V:

〈j(Z )X ,Y 〉 = 〈[X ,Y ],Z 〉 (X ,Y ∈ V).

Some historical remarks – Why the Chern-Rundconnection?

The Chern connection of a Finsler metric on a manifold wasoriginally introduced by S.-S. Chern (1943). The Chern connectionwas introduced again independently by H. Rund (1959, see alsoM. Anastasiei 1996). It was highlighted in the work of D. Bao andS.-S. Chern (1993), where the authors show the extraordinaryusefulness of the Chern connection in treating global problems ofFinsler geometry. In particular, the connection provides an easyway to compute the flag curvature of a Finsler metric.

Bao, David and Chern, S. S.

On a notable connection in Finsler geometry.

Houston J. Math., 19(1) 1993.

Some historical remarks – Why osculate?

The Chern-Rund connection when considered as a connection in afiber bundle over TM \ 0, it does not allow one to use directly thecoordinate-free global methods of modern differential geometryemployed in the study of Riemannian geometry. This can beovercome by using the osculating Riemannian metric associated toa Finsler metric introduced by A. Nazim in his Ph.D. thesis (1936).See also Moor (1952-85), Kozma (1992-1993).

H.-B. Rademacher.

Nonreversible Finsler metrics of positive flag curvature.

In D. Bao, R. Bryant, S.-S. Chern, and Z. Shen, editors, A Samplerof Riemann-Finsler Geometry, pages 261–302. CUP, 2004.

Finslerian toolsOsculating objects

Definition (Osculating Riemann metric)

〈X ,Y 〉W =1

2

∂2F 2(W + sX + tY )

∂s∂t

∣∣∣∣s,t=0

Definition (Osculating Cartan tensors)

CW (X ,Y ,Z ) =1

4

∂3F 2 (W + rX + sY + tZ )

∂r∂s∂t

∣∣∣∣r ,s,t=0⟨

C2W (X ,Y ),Z

⟩W

= CW (X ,Y ,Z ).

Osculating metricRanders case

Finslerian toolsChern–Rund connection

For left invariant vector fields

2⟨∇W

X Y ,Z⟩W

= 〈[X ,Y ],Z 〉W − 〈[Y ,Z ],X 〉W + 〈[Z ,X ],Y 〉W −

− 2CW (∇WX W ,Y ,Z )− 2CW (∇W

Y W ,Z ,X )+

+ 2CW (∇WZ W ,X ,Y ).

It is torsion-free:

∇WX Y −∇W

Y X − [X ,Y ] = 0,

and almost metric

X 〈Y ,Z 〉W =⟨∇W

X Y ,Z⟩W

+⟨

Y ,∇WX Z

⟩W

+ 2CW (∇WX W ,Y ,Z ).

Basic settingRanders metric on 3 dimensional Heisenberg group

Definition

Z = span Z be. Let (X ,Y ) be any basis of R2. Define[X ,Y ] = −[Y ,X ] = Z with all other brackets zero. The Liealgebra N = Z ⊕ R2 is the 3-dimensional Heisenberg algebra.Moreover, let 〈, 〉 denote the positive definite inner product on Nfor which (X ,Y ,Z ) is an orthonormal base.

For X0 ∈ N with property ‖X0‖ < 1 the function

f : N → R, X 7→ f (X ) =√〈X ,X 〉+ 〈X0,X 〉

defines a Minkowski functional on N , therefore it can be extendedto a left-invariant Randers type Finsler metric F on the Lie groupN by left translations.In this talk X0 = ξZ .

Focus on three dimensional Heisenberg group

Theorem

Let G be a connected 3-dimensional Lie group with left invariantmetric. Choose an orientation for the Lie algebra G so that thecross product is definied. The Lie bracket in this Lie algebra isrelated to the cross product by the formula [U,V ] = L(U × V ),where L is linear. G is unimodular iff L is self adjoint.

+ON-base: L(ei ) = λiei , where three eigenvalues are well-definedup to order.

Focus on three dimensional Heisenberg group

sgn Lie group description

+ + + SU(2) ∨ SO(3) compact, simple

+ +− SL(2,R) ∨ O(1, 2) noncompact, simple

+ + 0 E(2) solvable

+− 0 E(1, 1) solvable

+ 0 0 H(3) nilpotent

0 0 0 R ⊕ R ⊕ R commutative

[E1,E2] = E3, [E1,E3] = 0, [E2,E3] = 0 =⇒ [G, [G,G]] = 0

The Berwald-Moor frame

A. Moor.

Uber die Torsions- und Krummungsinvarianten derdreidimensionalen Finslerschen Raume.

Math. Nachr., 16:85–99, 1957.

Matsumoto, Makoto.

A theory of three-dimensional Finsler spaces in terms of scalars.

Demonstratio Math., 6:223–251, 1973.

The Berwald-Moor frame

E1 =1√

〈W ,W 〉WW

Definition (Cartan vector)

∀S ∈ N : 〈S ,CW 〉W = (trace C 2W )(S) = trace(U 7→ C 2

W (S ,U)).

〈W ,CW 〉W = 0, (+[CW ,W ] = 0 in our case).

T. Mestdag and V. Toth.

On the geometry of Randers manifolds.

Rep. Math. Phys., 50(2):167–193, 2002.

The Berwald-Moor frame

Proposition and Definition

W /∈ Z =⇒ CW 6= 0 and

E2 =1√

〈CW ,CW 〉WCW (W /∈ Z).

Turnout: The Chern – Rund connection in arbitraryHeisenberg group, if W ∈ Z

Levi-Civita∀X ,Y ∈ V; Z ,Z ∗ ∈ Z:

∇XY = 12 [X ,Y ]

∇XZ = −12 j(Z )X

∇ZZ ∗ = 0

Chern – Rund, W ∈ Z∀X ,Y ∈ V; Z ,Z ∗ ∈ Z:

. . .+ 12DW (X ,Y )

∇WX Z = −1

2 jW (Z )X

∇WZ Z ∗ = 0

〈jW (Z )X ,Y 〉W = 〈[X ,Y ],Z 〉W , (X ,Y ∈ V)

DW (X ,Y ) = C(jW (W )X ,Y ) + C(jW (W )Y ,X ) + jW (W )C(X ,Y )

A. Toth and Z. Kovacs.

On the geometry of two-step nilpotent groups with left invariantFinsler metrics.

Acta Math. Acad. Paedagog. Nyhazi. (N.S.), 24(1):155–168, 2008.

Turnout: The Chern – Rund connection in 3 dimensionalHeisenberg group, if W ∈ Z

Proposition

The local components of the Chern–Rund connection ∇Z w.r.t.base (X ,Y ,Z ) are

∇ZXX = 0, ∇Z

XY =1

2Z , ∇Z

Y X = −1

2Z , ∇Z

Y Y = 0,

∇ZZX = ∇Z

XZ = −ξ + 1

2Y , ∇Z

ZY = ∇ZY Z =

ξ + 1

2X ,

∇ZZZ = 0.

The Berwald-Moor frame (continued)

If W /∈ Z then CW 6= 0 and E2 = 1√〈CW ,CW 〉W

CW .

Proposition

E3 = W × E2 w.r.t. 〈, 〉,

[E3,E1] = 〈E2,Z 〉Z

[E2,E3] =(〈E2,E2〉 〈W ,Z 〉+ ξ 〈Z ,E2〉2

)Z .

Recall the Chern–Rund connection

For left invariant vector fields

2⟨∇W

X Y ,Z⟩W

= 〈[X ,Y ],Z 〉W − 〈[Y ,Z ],X 〉W + 〈[Z ,X ],Y 〉W −

− 2CW (∇WX W ,Y ,Z )− 2CW (∇W

Y W ,Z ,X )+

+ 2CW (∇WZ W ,X ,Y ).

2⟨∇W

X Y ,Z⟩W

= A(X ,Y ,Z ) + B(X ,Y ,Z ,∇W )

Our aim is to express the W -connection from the above equation.

The Chern – Rund connection (algorithm)

2⟨∇W

X Y ,Z⟩W

= A(X ,Y ,Z ) + B(X ,Y ,Z ,∇W )

1

2⟨∇W

W W ,Ei

⟩W

= A(W ,W ,Ei ) =⇒ ∇WW W

2 S ∈ {Ei}.

2⟨∇W

S W ,Ei

⟩W

= A(S ,W ,Ei )− 2CW (∇WW W ,Ei ,S) =⇒ ∇W

S W

3 Let S ,T ∈ {Ei}.

2⟨∇W

S T ,Ei

⟩W

= A(S ,T ,Ei )− 2CW (∇WS W ,T ,Ei )−

− 2CW (∇WT W ,Ei ,S) + 2CW (∇W

EiW ,S ,T ).

The Chern – Rund connection

Proposition

∇WE1

W = f1E3, ∇WE2

W = f2E3, ∇WE3

W = f3E2

∇WE1

E1 =f1

wE3, ∇W

E1E2 = ∇W

E2E1 =

f2

wE3, ∇W

E2E2 = f4E3,

∇WE3

E1 = ∇WE1

E3 + [E3,E1] =f3

wE2,

∇WE2

E3 = ∇WE3

E2 + [E2,E3] = − f2

wE1 + f5E2,

∇WE3

E3 = − f3

2〈X0,E2〉E3,

where f1, . . . , f5 are functions of w = ‖W ‖W .

Flag curvature

Flag:

a basepoint p

a flagpole W ∈ TpM

an edge U ∈ TpM (transverse to the pole)

Flag curvature: Π = span(U,W )

K (Π,W ) = K (Π) =〈R(U,W )W ,U〉W

〈U,U〉W 〈W ,W 〉W − 〈U,W 〉2W

Theorem

If W ,U ∈ span(X ,Y ) then K (Π) = ξ2−34 < 0

If W ∈ span(X ,Y ), U = Z then K (Π) = 1−ξ2

4 > 0

If W = Z , U ∈ span(X ,Y ), then K (Π) = 14 > 0

Bi-invariancy

Two-step nilpotent Lie groups endowed with a left invariantRiemannian metric cannot be also right invariant.

Theorem (Latifi 2010)

For arbitrary Lie groups: bi-invariant Randers metric is of Berwaldtype.

Some recent papers

E. Esrafilian and H. R. S. Moghaddam.

Flag curvature of invariant Randers metrics on homogeneousmanifolds.

J. Phys. A, 39(13):3319–3324, 2006.

D. Latifi.

Bi-invariant Randers metrics on Lie groups.

Publ. Math. Debrecen, 76(1-2):219–226, 2010.

H. R. S. Moghaddam.

On the Randers metrics on two-step homogeneous nilmanifolds ofdimension five,

Int. J. Geom. Methods Mod. Phys., 8(3):501–510, 2011.