On total null manifolds included in a real pseudo-euclidean space of signature (n, n 4- 1)

B y RADU ROSgA

Introduction. Let x : V n --> V "+v be the inclusion of an n-dimensional C~-manifold V n in a Riemannian or pseudo-Riemannian space V n+v the fundamenta l tensor of which is g. The reciprocal image x*g = g o x defines on V" a p roper Riemannian or pseudo-Riemannian s t ructure , if the r ank r of the map x is n (in t h a t case x is an immersion). I f the form x*g is degenera ted t hen V" is an improper or null submani/old of V "+~.

In t h a t case r < n and d = n - - r is called the de/ect of the null manifold V n (which is denoted by V~(d)). I n par t icu lar if d = n, V n is called a total null or null saturated [1] manifold and is denoted by V?n). As it is known, null submanifolds of a four dimensional space-t ime manifold, p lay in different respects an impor t an t role in Rela t iv i ty . In some pre- vious papers, [2], [3], [4], [5] the au thor has s tudied alone or in collabo- ra t ion wi th L. VA~HECK~. some types of null manifolds in an n-dimen- sional hyperbol ic space.

This paper is concerned wi th to ta l null real manifolds of dimension

n and codimension n A- 1 included in a real pseudo-eucl idean space ~ , , + 1 of inertia index n (i. e. o f s ignature (n, n ~- 1)). We recall t h a t a real C~-manifold V ~+1 of odd dimension 2n + 1 is said to be almost cosymplectie i f a 1-form O e A 1 (V ~'+1) and a 2-forme H e Ae(V 2n+~) of r ank 2n(Av{V2~+l): modul of all p-forms on V 2n+x) are given such t h a t

t h e y sat isfy O ^ (A"II) ~ 0 (see also [6], p. 218). On the o ther hand according to P. LIBERMANI~ [7], p. 64, to a qua-

drat ic form of signature (n, n) is associated a real quadra t ic form called parahermitian and by complexificat ion of a cosymplect ic s t ruc ture J . BovzoN [8] p. 412 has defined an almost cohermetian structure. Mak- ing use of these notions we have endowed the space ~ ,"+~ wi th a geo- metr ical s t ruc ture (subordinated to an almost cosymplect ie s t ruc ture

C) t e rmed a paracohermetian structure and denoted by ~vc . The un i t a ry ver t ical vec tor field ~ of ~r~ (~ is eolinear to the field E

of Reeb associated wi th C, i.e. is defined b y i~O = 1, i~H = 0) plays

a dist inguished role wi th respect to the considered inclusion x. L inked to ~ is also a set of connect ion forms which are called the characteristic

10 Radu Ros~a

connection/orms of the principal bundle ~ (~n. ~+~) over ~ . These forms are all conformal to a 1-form oJ, called the characteristic ]orm associated with x. The form o) is completely integrable and the only non vanishing second quadratic fundamental form, associated with x, is eonformal to (w)2. I t should be remarked that this set of properties has been already found by the author in the case of what he has defined as being a pseudo- null immersion of submanifolds in a Minkowski n-dimensional space [9], [10]. I f V~n~ is of even dimension i. e. n ---- 2 m a certain almost sympleetio structure [V~',), Q] is considered. Different properties involving Q, o~ and some vector fields on V[,) are studied.

1. Let ~","+~ be a real pseudo-euclidean C~ of signature (n ,n + 1) and let ~(~n, ,+1) be the tangent space at each point p E ~","+~. One may write

(1) ~ ( ~ , " + ~ ) : ~ @ D~

where ~ and D~ are a parahermitian vector space [7] p. 64 of dimension 2n and a space-like line, respectively. Let S~ and 27"" be the time-like and space-like space respectively, which define an involutary automor- phism 1I on ~ " , such that U z ---- Jr 1. The field D~ is orthogonal to ~ " and I I ( X ) ~ 0, for any field X e D , . Let ~i, ~i* (i----1, 2 , . . . , n ; i* -~ i -~ n) and ~----~,+1 be the null real vectors which define the canonical basis of ~ " , and the space-like vector of D~, respectively.

I f e~ e S~ and e~. E 27p*- is an orthonormal basis, then by means of the transformation

(2) ~ _ e, + Ue,.

it is readily proved that the frames (p, ~A; A, B : 1 . . . 2n + 1} are normed; that is

(3) (~,, ~A) ---- 0, (~r ~B} = 0, (~,, ~,.} ---- 1, (~, ~} ---- - - 1 ; A ~ i*, B 4 i

I f {eTA} is the associated co/rame of (p, ~A}, then the line element dp of ~",~+~ is

(4) dp -~ ~a | ~A.

By means of (3) one finds that the metric ~ of @n,,+~ in terms 07A is given by

(5) �9 -~ (dp, dp) -~ X,~'07'" - - (07)2; 07 ---- 072,+1.

The quadratic form ~ ----27~07i07 ~* and 07 are the quadratic parahermitian form of ~ " and the associated covector of ~, respectively. The form and its associated 2-form lIZ~cT~ ̂ 07i* are invariant order the para-

On total null manifolds included in a real pseudo-euclidean etc. 11

unitary group D n [1] (this group is isomorphic to the linear real group L n [7] p. 65). By considerations analogous to tha t made in [8] p. 412 we may say tha t {r ~, D~} defines a para-cohermitian structure ~ c on @=,n+l. Let ~(@n,n+i): U{p, ~A} be the principal bundle of the frames {P, ~A} over @n, ~+1 and 07~ = l~c07 c the connection forms on ~ (@", n+l). The space @ is structured by the connection

(6) ? I~A ---- 07A B @ ~B

and by virtue of (3) one gets

07~. = 0, 0 7 ~ * = 0, 071~++I = 0,

(7) 07~ + 07{: = 0, 07~, + 07{, = 0, - i * - i ~T,2n+l 072n+1 + ~2.+~ = O, ~.+1 + ~ , = O.

Since the connection V is torsion an curvature free, both groups of structural equations (of E. CAI{TA~) are

(8) d ^ 07~ = 07B ^ 07~

(83 d ^ 07~ = 07~ ^ 07~

2. Consider now the inclusion x: V ~ -+ (~n,~+~ of a p-dimensional C+-manifold in ~","+~. On says that V ~ is total null, (see Introduction) if the defect of the manifold is equal to its dimension. Referring to the metric (3) we readily see tha t max p = n (see also E. CARTA~X [11], p. 10). In the following we shall be concerned with total null manifolds V"C~ ~,~+~ of maximal dimension n (denoted by V~,)). Such a submanifold of ~", n+~ may be considered an integral manifold of the system

( 9 ) 07~* = 0 , 07 = 0 .

V ~ From (9) and (6) it follows tha t the tangent space ~ ( ( , > ) at x(p)~ V~n> is sel/-orthogonal [12], p. 76. Therefore if we denote by ~ V n ( (,)) and ~v:(V~,)) the total normal space at x(p) and the supplementary space, respectively, we have

n ~:~ (V~>) ~ ~ (~-,"+~)/~;~ (V<~))

(10) ~:v l V<~) = ~(V('~) Q D~

The covariant di~erential of any vector field X e ~ (V~) ) , induced by x, can be decomposed into two parts

vx = (vx)n + (vx)~

where (V X)n is the normal part and (V X)~ the supplementary part over V[,) in @n,"+x. I f (VX)~ = 0 identically, we say [13] that X is parallel

12 Radu Ros~a

in the normal bundle u ~ s V~n). Denote by ~i, ~ the values of o7 a, 5a respectively, induced by x. We shall call (V ~i, ~ ) : ~* the tangential torsion /orms and (V ~2n+l, gi> : - - 0 ~ n+l the characteristic connection /orms, associated with x.

Equations (6) show that the necessary and sufficient conditions that all tangential vectors ~ be parallel in the supplementary bundle is ~* ---- O.

Next by virtue of (10), the n + 1 second fundamental forms asso- ciated with x, are

(11) ~ , = --<dx(p), Vg,> = --x~* (11') 9---- --<dx(p), Vg) ---- 0~cr n+l

and with the help of (8), exterior differentiation of (9) gives

(12) cq ---- 0 ~ 0 i O.

I t follows that the space Ov(V~*.)) (the normal subspace of which the V ~ second fundamental forms are null) coincides with ~ ( ~n)) and that

the first normal space coincides with D~. Hence we may formulate

Theorem. Being given the inclusion x o/ a total null manitold V(~) in ~,'~+~, we have the ]ollowing properties:

(i) all tangential torsion/orms associated with x are null,

(ii) Chern's arithmetic invariant is 2, and the first normal space associated with x coincides with the line D~ o] the structure ~ o ,

(iii) all the tangential vectors ~i are parallel in the supplementary bundle.

3. Exterior differentiation of (12), gives with the help of (8')

(13) c~ ~+1 ---- a~(akc~k); ai, ~ e ~3(V~,~)).

We shall call 2(ak~ k) eAI(V~,~) the characteristic 1-/orms associated with x, and taking the exterior differentials we get

(14) akd ^ co = (a~o~L--da~) ^ co.

Hence ~ is completely integrable, and this property is similar to one of the basic properties of pseudo-null immersions [9], [10]. On the other hand the Lipschitz-Killing curvature [14], corresponding to the non null normal vector ~ e D~, being

K(p, ~) --~ det [~(p, ~)[

one gets from (11') and (13), K(p, ~) ---- 0. Remarking that the Lipschitz-Kflling curvatures corresponding to

the null vectors ~, are trivially zero, we have

On total null manifolds included in a real pseudo-euclidean etc. 13

Theorem. The characteristic connections /orms associated with the in- clusion x: V'~n~ --> ~n, ,+x are all con/ormal to a 1-lotto (characteristic form) which is completely integrable and all the Lipschitz-Kill ing curvatures asso- ciated with x are null.

4. We shall call the vector field

the characteristic field associated with the inclusion x. B y (6) we get

(15) V X~ Z~ (da~ - - a~ ~ ) ~* ~

where

(15') Yi~ = a ~ - - ak~i.

F rom (14) it is seen tha t the necessary and sufficient condit ion tha t X0 be parallel in the supplementary bundle is tha t ~ be closed. Next b y (8') one finds tha t the sys tem obta ined from

(16) d A ~---- 0 ~ d a i - - a ~ a ~ - ~ 0

b y exterior differentiation, is an algebraic corollary of (16). We conclude t ha t the sys tem (16) is completely integrable and therefore the manifolds V~n) for which (16) holds (denoted b y V~(~)) depend on n arbitrary con- stants.

5. Denote now b y V the volume element of V~n). As is known [15] p. 6, r/ defines an S L ( n ; R)-s t ructure on V[~) (the cross section of the bundle ~(V~'n))/SL (n; R) are in one-to-one correspondence with the volume elements of Vl*n~). One finds tha t the fields Y~k given b y (15'), are related to ~ b y an (n - - 1)-linear map ofYik inA n (V~)). I f considering the set YI~ (7" ---- n, n - 1 , . . . , 3, 2), we have

iy1 ~ , . . . , i~1~ V = ( - - a l ) n-2 ~/2

(iy: inner product of forms with respect to a vector field Y). Next , if is closed, we get from (15) and (16)

(17) V Y~ = (~I + ~ ) Y ~ + Z ~ ( ~ Y ~ - - a~'Yk~); r=4=i,k

and this proves tha t the fields Y~k are null of infinite type [16]. Moreover, since Y i k e ~ (V~'n~) b y (16) we obtain ~y,,V----(divYi~)V (~y----Lie der ivate of forms with respect to a vector field Y).

Hence, if ~ is closed, the fields u are divergence/tee on V,~n). In o~her words each Yik is an infinitesimal automorphism of the SL (n; R)-s t ructure under consideration. In the case V" (n) is of even dimension, i. e. n = 2 m,

V" let us introduce an almost symplectic (local) s t ructure [ in), 12] having

14 Radu Rosga

for/undamental bivector aiak (ai ak is an invariant of the sympleetic group Sp (m, R)). By definition, we have

{18) ,.('d~ ~ ~ i < k a i a k o ~ i ^ o~ k

and for any pair (Y~k,Yh~) such that i ~= h, l; k =~ h, l, we get

Thus, the fields Y~k may be assembled in m (with respect to f2~) involutive subspace sof ~(V~nl). Hence

Theorem. With every inclusion x: [V?n),co] ~ ~.,.+1 there is associaded (2)

null vector fields Y ~ ~ 7~(V'~.I) such that n - - 1 contractions by Y~k o/ the volume element ~l o] V'~n) gives a ]orm con/ormal to o). Furthermore: (i) / / ~o is closed then the fields Yik are null o/ infinite type and each o/ them is an infinitesimal automorphismen o/the SL (n; R)-structure on V'~n~. (ii) I / one considers the almost symplectic structure (18) then the fields Yik, define m involutive maximal subspaces o/ ~(V'~)) .

6. Being given on V[~) the null vector field

(19) N ---- 27,#~i; #, ~ ~)(V~'.,)

we shall inquire under what conditions there does exist a nowhere vanishing N such that

(19') VN = 0.

According to K. YAz~o and B. Y. CEEN [17] and remarking that N C ~:~ (V[.)) n ~(V~n)), we shall say that if (19') is satisfied then the inclusion x is not substantial. By (6) we find that (19') leads to the condition

(20) Z i / ~ i a i = 0

and to the pfaffian system

(21) dm + 2 ~ / ~ = 0.

By (8) one finds that the necessary and sufficient condition that (21) be completely integrable is that (20) holds. Thus any inclusion x: Vn(n)-+ ~,,~+1 is not substantial.

Moreover by a straightforward cMculation it can be seen that if (20) holds then the map 9~(p) : V~.)-+ ~.,.+1 defined by

9~(p) = - x ( p ) - - N

is spherical [18] and with vanishing ray. Next consider the map 2r : V~.)-+ ~","+a whose generator is the field ~ ~ D~.

On total null manifolds included in a real pseudo-euclidean etc. 15

We m a y write

(22) 9~(p) ---- x(p) -1- flt~; t3 e ~(V~'n))

and taking the covariant differential of 9~(p) we get by (6)

(23) vg~(p) -- ( ~ -4- fla~n+a) | ~ + dfl @ ~ - - f l o ) @ Xc.

As is known, for the integral manifolds V~ corresponding to the line element (23) to exist, it is necessary and sufficient t h a t

(24) d h V 9.1~ (p) ---- 0.

B y (8) and (8') we find tha t (24) implies

( 2 5 ) =

and so the metric (ds~) 2 of V~ is expressed by

(26) (ds~) 2 ---- ---{~ q- 2fl(1/2 q- f lXia, / , )}(w) ~.

From the above representation of (ds~) 2 we deduce t h a t V~ is an n-dimensional null manifold of de/ect n - 1.

In part icular i f / , and ~ are such t h a t

Q2 + 2fl(1/2 q- f l S , , a , / , ) = 0

the manifold V~ under consideration is of the same definition as V~',).

Hence we have

Theorem. A n y inclusion x : V'~n) --> ~",'~+~ is not substantial and / tom a given V~,,~ one m a y derive c~ null mani/olds o/ de/ect n - 1.

8. Le t 2 be the isomorphism which defines for each vector X E ~ (V~)) a 1-form o~x eAI(V'~,~) such tha t o~x -~ ixt2. As is known we m a y set X = 9~ -1 (~x) and if the 1-form ~x is the characteristic form o~ of VT~) we shall write

=

Referring to (18) one finds

(27) X~ ---- 1 2 : , ( - - 1)/-1 ~ial . . , a i '*" a2,

where the roof indicates the missing term and A = a l , �9 �9 a ~ is P/af f ' s agregate of maximal order with respect to D~.

On the other hand the symplectic ad]oint ~ of w is according to the general definition [7], p. 66 given by

1 (28) o) = (m-- 1-----~ t ~ ^ (A~-~2~).

16 Radu Rosga

But from (18) we deduce

(29) A~-1~2~ = ( m - - 1 ) ! ~aQ 1 . . . . o~m_2aol " ' " ao~n_20~~ ^ " ' " ^ Ot ~

and introducing in (28) we get

* a01 m--106Q1 (30) O~ ~---d~aql...aO2m_laQ1... ^ . . . ^ 0~ Q2mxx.

Now if we contract the volume element ~ by X~, we obtain

(31)

But

(32)

and referring to (31) and (28) we finally have

(div, X~)~ = - - ( m - - 1) ^ ~ -{- (m --2)IA ^ dA(A'~-IT2r)

and we may formulate

Theorem. Being given the inclusion x : V~'n~ -+ @,,,+1 (n = 2m) let oJ and Q~, be the characteristic/orm and the almost symplectic ]orm associated with x respectively and A P/a~'s agregate o/ maximal order with re-

spect to Q~,. I[ dA is a linear divisor o/ the symplectic adjoint & o/ o~, then the necessary and su/ficient condition that the field 2 -l( oJ) be divergence ]ree is that T2v be symplectic or con/ormal sympIectic.

9. Since the manifold V~n) is total null, one may consider as in [2] the associated linear element of dx(p). We shall denote this element by dp~ and according to [2] dpa is given by

(33) dpa = ~' | ~,* E~#, (V~',,).

By means of (33), the main vectorial ( 2 m - 1)-form Ha [2] associated with x is expressed by

(34) Ha --(2m--1)!,,l],, d p a " " dpa, ~ x ' " ~ , , [ [ , , 2m--- 1

where the symbol i',[[ii ',',nll denotes the combined operator of exterior product (^) of forms and exterior product ( • ) of vectors. Exterior dif- ferentiation of (34) gives by (6) and (33)

(35) d ^ H~ = {Z~(div ~, + traee(lli))~, + trace (li~+i) ~}~}

and the vector field

(35') H = Z,(div ~i + trace (llj))~, + trace (li~ +1)

On tota l null manifolds included in a real pseudo-euclidean etc. 17

w h i c h is i n d e p e n d e n t o f t h e cho ice o f t h e bas i s {~A} is de f i ne d as t h e

mean curvature vector a s s o c i a t e d w i t h x.

I f H v a n i s h e s i d e n t i c a l l y w e s a y t h a t t h e i nc lu s ion x is minimal. C o n s i d e r n o w t h e a d j o i n t & o f ~ w i t h r e s p e c t t o t h e v o l u m e e l e m e n t

7. O n e h a s

(36) & = X ~ ( - - 1 ) ~ - l a ~ 1 A �9 �9 �9 A &~ A �9 �9 �9 h 0 r n

a n d t a k i n g a c c o u n t o f (35') , e x t e r i o r d i f f e r e n t i a t i o n o f (36) g ives

d ^ ~ = <Xo, H)~/

a n d so we h a v e

T h e o r e m . I/ the characteristic [orm ~ associated with the inclusion x : [V~,~, Q ~ ] - > ~ , , + 1 (n = 2m) is closed, then the necessary and su/- ficient condition that co be harmonic, for any characteristiv field X~, is that the inclusion x be minimal.

R e f e r e n c e s

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[2] R. RoscA et L. VA~CH~CKE, Sur les varidtd to ta lement isotropes de codimen- sion g6oddsique 1 incluses dans une varlet6 lorentzienne de dimension paire, C. R. Paris. t- 275, 1972, serie A.

[3] R. RoscA, On null hypersurfaees of a Lorentzian manifold. Tensor N. S. vol. 23 (1972).

[4] R. RoscA et L. VA~HECKE, Sur les hypersurfaces isotropes V incluses dans une vari6te lorentzienne et admet t an t un champ eoncourant dans le plan lorentien associ6e. C. R. Acad. Se. Paris, t . 274, p. 1635--1638, 1972.

[5] R. RoscA, Surfaces isotropes de ddfaut 1 ineluses dans une varigt6 lorentziene Acad. Royale de Belgique, 5 ~ m e sdrie t. LVII , 1971.

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[7] P. LIBERMA-~, Sur le probl~me d'~quivalence de certaines structures in- fmit~simales. Annali di Matematica, t . 36 1954, p. 27--100.

[8] J. BovzoN, Structures presque cohermitiennes. C. R. Sc. Paris t. 258 (1964) groupe I I , p. 412--414.

[9] R. RoS, CA, Les vari6tds pseudo-isotropes dans un espace temps de Minkowski, (~. R. Sc. Paris, t . 270 (1970) s~rie A. p. 1071--1073.

[10] R. RoscA, Les hypersurfaces pseudo-isotropes dans un espace de Minkowski, Acad. Royale de Belgique 5-eme sdrie, t . LVI, 1970--1974.

[11] E. CARTAN, The theory of spinors. Hermann 1966. [12] J . M. SOURIAU, Structure des systbmes dynamiques (p. 76). Dunod, Paris 1970. [13] K. YAHOO and B. Y. CHE~, Minimal submanifolds of a higher dimensional

sphere. Tensor N. S. vol. 22 (1971) p. 369. [14] B. Y. C H ~ , On the to ta l absolute curvature of manifolds immersed in Rie-

mannian manifolds I I . KSdai Math. Sem. Rep. vol. 22 no. 1, pp. 89--97.

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18 Radu Ros~a

[15] S. KOBAYASHI, Transformation groups in differential Geometry. Springer Verlag, Berlin 1972.

[16] E. BOMPIA~I, Intorno alle varieta isotrope. Ann. Matem. Serie IV. 20, 1940. [17] K. Ym'~o and B. Y. CHEN, On the concurrent vector fields of immersed mani-

folds, K5dai Math. Sem. Rep. Vol. 23, pp. 343--250, 1971. [18] B. Y. C~EN and K. YANO, Pseudo-umbilical submanlfolds in a Riemannian

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