some properties of lightlike submanifolds of semi-riemannian ...

DEMONSTRATE MATHEMATICA Vol. XLIII No 3 2010

Rakesh Kumar, Rachna Rani, R. K. Nagaich

SOME PROPERTIES OF LIGHTLIKE SUBMANIFOLDS OF SEMI-RIEMANNIAN MANIFOLDS

Abstrac t . We initially obtain various relations and then establish necessary and sufficient condition for the integrability of screen distribution of a lightlike submanifold. We also establish necessary and sufficient condition for a lightlike submanifold to be totally geodesic.

1. Introduction In the generalization from Riemannian to semi-Riemannian manifolds,

the induced metric may be degenerate (lightlike) therefore there is a natural existence of lightlike submanifolds and for which the local and global geom-etry is completely different than non-degenerate case. In lightlike case the standard text book definitions do not make sense and one fails to use the theory of non-degenerate geometry in the usual way. The primary difference between the lightlike submanifolds and the non-degenerate submanifolds is that in the first case, the normal vector bundle intersects with the tangent bundle. Thus, the study of lightlike submanifolds becomes more difficult and different from the study of non-degenerate submanifolds. Moreover, the geometry of lightlike submanifolds is used in mathematical physics, in par-ticular, in general relativity since lightlike submanifolds produce models of different types of horizons (event horizons, Cauchy's horizons, Kruskal's hori-zons). The universe can be represented as a four dimensional submanifold embedded in a (4 + n)-dimensional spacetime manifold. Lightlike hyper-surfaces are also studied in the theory of electromagnetism [1], Thus, large number of applications but limited information available, motivated us to do research on this subject matter. Kupeli [6] and Bejancu-Duggal [1] developed the general theory of degenerate (lightlike) submanifolds. They constructed a transversal vector bundle of lightlike submanifold and investigated vari-

2000 Mathematics Subject Classification: 53B25, 53B40, 53C50. Key words and phrases: lightlike submanifolds, totally geodesic lightlike submanifolds.

692 R. Kumar, R. Rani, R. K. Nagaich

ous properties of these manifolds. This paper is organized as, in Section 2, we recall some basic definitions and formulas for lightlike submanifolds of semi-Riemannian manifolds. In Section 3, we obtain various relations and as an application establish necessary and sufficient condition for the integra-bility of screen distribution. Finally, a necessary and sufficient condition for a lightlike submanifold to be a totally geodesic is discussed.

2. Lightlike submanifolds Let (M, g) be a real (m + n)-dimensional semi-Riemannian manifold of

constant index q such that m,n > 1, 1 < q < m + n — 1 and (M,g ) be an m-dimensional submanifold of M and g the induced metric of g on M. If g is degenerate on the tangent bundle TM of M then M is called a lightlike submanifold of M. For a degenerate metric g on M (1) T M 1 = U{u e TXM : g(u,v) = 0, Vi; £ TxM,x e M}, is a degenerate n-dimensional subspace of TXM. Thus, both TXM and TXM^ are degenerate orthogonal subspaces but no longer complementary. In this case, there exists a subspace RadTxM = TXM n TxM1- which is known as radical (null) subspace. If the mapping (2) Rad TM : x e M —> Rad TXM, defines a smooth distribution on M of rank r > 0 then the submanifold M of M is called r-lightlike submanifold and RadTM is called the radical distribution on M. Following are four possible cases:

(i) M is r-lightlike submanifold if r < min{m,n}, (ii) M is Coisotropic submanifold if r = n < m,

(iii) M is Isotropic submanifold if r = m < n, (iv) M is Totally lightlike submanifold if r = m = n.

For case (i), there exists a non-degenerate screen distribution S(TM) which is complementary vector subbundle to RadTM in TM. Therefore (3) TM = Rad TM±S(TM), S,(TM_l) is a complementary vector subbundle to RadTM in TM±. Let tr(TM) and ltr(TM) be complementary (but not orthogonal) vector bundles to TM in TM \M and to RadTM in ^ ( T M 1 ) 1 respectively. Then we have (4) tr(TM) = ltr(TM)±S(TM±).

(5) TM\M = TM ® tr{TM) = (Rad TM © ltr(TM))±S(TM)±S(TM±).

For case (ii), we have Rad(TM) = TM1 therefore S(TML) = {0} and tr(TM) = ltr(TM) and (6) TM\M = (TM1- © ltr(TM))-LS(TM).

Some properties of lightlike submanifolds of semi-Riemannian manifolds 693

For case (iii), we have RadTM = TM then S{TM) = {0} therefore

(7) TM\M = CTM © ltr(TM))±S(TM±).

For case (iv), we have RadTM = TM = TM1 then S(TM) = S(TM±) = {0} therefore

(8) TM | M = TM © ltr(TM).

Let u be a local coordinate neighborhood of M and consider the lo-cal quasi-orthonormal fields of frames of M along M, on u as { f i , . . . , £ r , Wr+1, ...,Wn,Ni,...,Nr, Xr+1,..., Xm}, where . {Nu..., Nr} are local lightlike bases of T(RadTM\u), T(ltr(TM)\u) and {W r +i , . . . , Wn}, {Xr+i,..., Xm} are local orthonormal bases of T(S(TM1-)\U) and r ( S ( T M ) \ U ) , respectively.

On the existence of above local quasi-orthonormal field of frames of M along M we have

T H E O R E M . Let (M,g,S(TM),S(TM-L)) be an r-lightlike submanifold of a semi-Riemannian manifold (M,g). Then there exists a complementary vec-tor bundle ltr(TM) of RadTM in S(TM-L)1 and a basis of T(ltr(TM)\u) consisting of smooth section {^Vj} of S'(TM"L)"L|U where u is a coordinate neighborhood of M, such that

(9) g(Ni,£j) = 8ij,g(Ni,Nj) = 0,

where ... ,£r} is a lightlike basis ofr(Rad(TM)).

Define locally r-differential 1-forms {rji} on T(TM)) by

(10) rH(X) = g(X,Ni),VX€T(TM).

Let P be the projection of TM on S(TM) with respect to (3). Then T

(11) X = PX + i=i

for every X <E T{TM). Let V be the Levi-Civita connection on M. Then, according to the

decomposition (5), the Gauss and Weingarten formulas are given by (12) VXY = VxY + h(X,Y), \/X,Y 6 T(TM), (13) VXV = -AVX + VjcV, VX g T(TM), V e T(tr(TM)),

where {VxY,AvX} and {h(X, Y), V ^ F } belongs to T(TM), T(tr(TM)), respectively. Here V is a torsion-free linear connection on M, h is a sym-metric bilinear form on T(TM) which is called second fundamental form, Ay is linear operators on M, known as shape operator.


According to (4), consider the projection morphisms L and S of tr{TM) on ltr(TM) and S{TMr) respectively. Then (12) and (13) become

(14) VxY = VxY + hl(X,Y) + hs(X,Y),

(15) WxV = -AvX + DlxV + Ds

xV, where we put hl(X,Y) = L(h(X,Y)),hs(X,Y) = S(h(X,Y)),Dl

xV = L{Vj^V),Ds

xV = S(V^F). As hl and hs are F(ltr{TM))-valued and r(S,(TMJ-)-valued respectively,

therefore we call them the lightlike second fundamental form and the screen second fundamental form on M. In particular

(16) VXN =-ANX+ \7lxN + DS(X,N),

(17) VXW =-AWX+ VSXW+ Dl(X,W),

where X G F(TM), N G F(ltr(TM)) and W G T(5(TM±)) . Now, if M is either co-isotropic or totally lightlike (5(TMX) = 0), then

(14) and (15) become

(18) VxY = VxY + hl{X,Y),

(19) \7XN = -AnX + DlxN.

Using (14)—(19), (5)-(8), we obtain

(20) g(hs(X, Y),W) + g(Y, Dl(X, W)) = g(AwX, Y),

(21) g(hl(X, Y), 0 + g(Y, hl(X, 0) + g(Y, V x 0 = 0, (22) g{ANX, N') + g(N, AN,X) = 0,

for any £ G F(RadTM), W G T(5(TMX)) and N, N' G F(ltr(TM)). Now, we consider the decomposition (3), we can write

(23) V x P y = V*XPY + h* (X, PY), (24) = —A*^X + for any X,Y G F(TM) and £ G T(RadTM), where {S7*xPY,A\X} and {h*(X, PY), V ^ } belong to F{S(TM)) and T(RadTM) respectively. Here V* and V^f" are linear connections on S(TM) and Rad TM respectively. By using (14)—(15), (23)-(24) we obtain

(25) g(hl(X,PY),Z)=g(AlX,PY), (26) g(h*(X, PY), N) = g(ANX, PY), for any X,Y G F{TM) and £ G F(RadTM) and N G F(ltr(TM)). Replace Y by £ in (21), we have

(27) g(hl(X,£),0 = 0 , V I 6 FTM.


Replace X by £ in (25) and then using (27) we obtain

(28) Atf = 0.

3. Properties of lightlike submanifolds Consider a coordinate neighborhood u of M and {iVj, Wa} be the basis of

T(tr(TM)\u) where Ni € T{ltr{TM)\u) and Wa e r (5 (TM ± )| u ) then (14), (16) and (17) become

r n

( 2 9 ) V x Y ^ V x Y + J2h[(X,Y)Ni+ £ hsa(X,Y)Wa, i=1 a=r+1

where hl{X,Y) = £ T = 1 h[(X,Y)Ni this implies

( 3 0 ) h[(X,Y) = g(hl(X,Y),&),

and hs(X,Y) = E L r + i h'a(X,Y)Wa this implies

( 3 1 ) eah'a{X,Y) = g{h'{X,Y),Wa),

where ea is signature of Wa. r n

( 3 2 ) VxNi = -ANiX + Y i p i j ( X ) N j + £ ria(X)Wa,

j=1 a=r+1

where VlxNi = Pij(X)Nj this implies

( 3 3 ) Pij(X)=g(VxNi,Sj),

and DS(X, Ni) = Y,l=r+iria(X)Wa this implies

( 3 4 ) eaTia = g(Ds(X,Ni),Wa), r 7i

( 3 5 ) VxWa = -AWaX + J2^i(X)Ni+ e<*p{X)Wp,

i=1 0=r+1

where Dl(X, Wa) = £ [ = 1 vai{X)Ni this implies

( 3 6 ) vai(X)=g(Dl(X,Wa),(;i),

and VsxWa = J2p=r+i Oap{X)W0 this implies

(37) el3eal3(X)=g(VsxWa,Wf3). (17) can also be written as

r n

( 3 8 ) VxWa = -AWaX + ^Dli(X,Wa)Ni+ £ 0a0(X)W0,

i=l /3=r+l where

( 3 9 ) Dl(X, Wa) = g(Dl(X, Wa),&.

696 E. Kumar, R. Rani, R. K. Nagaich

Using (39) in (20) we obtain r

(40) eah'a(X, Y) + J2 D\{X, Wa)Vi(Y) = g(AWaX, Y).

i=1

Replace Y = in (40), we obtain

(41) Dlj(X,Wa) = -eah8a(X,Zj).

Now from (29) and taking account of (9) we have

(42) hli(X,Y)=g(VxY,Zi),

replace Y by f j in (42) and obtain

(43) h\{X,ij) + hlj{X,ii) = 0,

since V is metric connection so as a consequence of (43) we obtain

(44) h\{X,ii ) = 0.

From (22) it is also clear that

(45) g(ANiX,Ni) = 0.

By the basis { £ } of F(RadTM\u), (23) and (24) become r

(46) VXPY = V*XPY + Y, Ei(X, PY)b,

i=1

where

(47) Ei(X,PY) = g(h*(X,PY),Ni),

and r

(48) VxS = - A l X + J 2 u i j ( X K j , 3=1

where

(49) uij(X) = g(V*x^i,Nj).

Using (25) and (47) we have

(50) Ei(X,PY) = g(ANiX,PY),

from (26) we have

(51) hi(X,PY)=g(AlX,PY).

Using (14), (16), (24), (33) and (49) we have

(52) Uij(X) = -pji(X) + gfc, ANjX).


Hence (48) becomes r

(53) V x t ^ - A l X - Y . P j i W Z i -3 = 1

From (43) and (51), we get

(54) Afct = 0.

With the help of above equations, now we will investigate the integrability of the screen distribution. Since V is a torsion-free linear connection, by using (11), (23) and (53),we obtain

r

(55) [X, y] = V*XPY - \7*yPX + Y^h(x)AlY - n(Y)Aix] i=1

r

+ FY) - Ei(Y, P X ) + X(Vi(Y)) -i=1

r r r

1=1 j=l j=1 r r

j=l 3=1

Taking the scalar product with Nk, k = 1, . . . , r, we obtain

( 5 6 ) g([X, Y ] , N k ) = Ek(X, PY) - Ek(Y, P X ) + X(Vk(Y)) - Y ( V k ( X ) ) r

+ J2{Vi(X)Pki(Y) - V i ( Y ) P k i ( X ) } ¿=1

r

+ ANkX)m(Y) - g(Zi, ANkY)TH(X)}, i=1

or

( 5 7 ) g([X,Y},Nk) - X { m ( Y ) ) + Y ( V k ( X ) )

= E k ( X , P Y ) - E k ( Y , P X ) r

+ ^ 2 { V i ( X ) p k i ( Y ) - V i ( Y ) p k i ( X ) } i=1

T

+ £{<?(&, ANkX)Vi(Y) - g f a ANkY)Vi(X)},


this gives

(58) 2dr¡k{X, Y) = Ek(Y, PX) - Ek(X, PY) T

+ J2iVi(Y)pki(X) - Vi(X)pki(Y)} i=1 r

+ ^{g(Zi,ANkY)rH(X)-g(Zi,ANkX)rH(Y)}, i=1

using (10) and (11) in (58), we obtain

(59) 2d V k (PX, PY) = Ek(PY, PX) - Ek(PX, PY),

or

(60) rjk([PX, PY]) = Ek(PY, PX) - Ek(PX, PY).

Thus from (60), we have the following theorem

T H E O R E M . Let (M,g,S(TM),S(TM-i)) be an r-lightlike submanifold of a semi-Riemannian manifold (M,g). Then, the screen distribution S(TM) is integrable if and only if t]i, i = 1 , . . . , r are closed on S(TM).

In the case M is coisotropic submanifold, (55) becomes n

(61) [X,Y] = V*XPY - V*YPX + Y^[VÍ(X)AIY - rn(Y)A^.X] i=1

n + PY) - Ei{Y, PX) + X(th(Y)) - y

i=1 n n

+ E EfaWM^' - r¡i{y)Pji{X)íj}-i=i j = i

C O R O L L A R Y . Let M be a coisotropic submanifold of codimension n of semi-Riemannian manifold (M,g). Then, the screen distribution S(TM) is integrable if and only if r¡i, i = 1 , . . . ,n are closed on S(TM).

Let M be a lightlike submanifold of semi-Riemannian manifold (M ,g). Denote R and R the curvature tensors of V and V respectively then by straightforward, we have

T

(62) R(X, Y)Z = R(X, Y)Z + Z) - (Vyh\)(X, Z)]Ni i=1

n + E l(Vxhsa)(Y,Z)-(VYhsa)(X,Z)]Wa

a=r+1


r + Z)ANIY - h[(Y, Z)ANIX]

i=1 n

+ ^[hsa(X,Z)AWaY-hsa(Y,Z)AWaX] a=r+1

r r

+ E z w * ) - h i ( x > ^ ( y m ¿=i j=i

r n

+ E E M y > - hi(x> Z)ria(Y)]Wa ¿=1 a=r+1

n r + E Y l K f r Z ^ d W - h i i X ^ f r a i i Y K N i

Q=r+1i=l n n

+ E Q=r+1 /3=r+l

If M is coisotropic submanifold of M then (62) becomes n

(63) R(X, Y)Z = R(X, Y)Z + EK^X1^Z) ~ Z ) ] ^ i=l

n + E l t o Z)ANiY - h{(Y, Z)ANiX}

i=1 n n

+ E - to ^ K ( ^ ) ] ^ - . ¿=i j=i

therefore form (62) and (63) we have the following theorem.

T H E O R E M . Let (M,g ) 6e a lightlike submanifold or coisotropic submanifold of (M,g) of codimension n. If M is totally geodesic then

(64) R(X,Y) = R{X,Y)

for any X,Y e T(TM).

D E F I N I T I O N . A distribution D on M is called a killing distribution if Lxg = 0 for each vector field X £ T(D), L being the Lie derivative.

Now

(65) g(hl(X, Y),N) = g(VxY, N) = -g(Y, VXN) = -g(Y,[X,N})-g(Y,VNX)


= g(Y, [N, X]) - Ng(Y, X) + g(VNY, X) = -Ng(X, Y) + g(Y, [N, X}) + g([N, Y],X) — g(N, VYX) = -(LNg)(X,Y)-g(N,VyX),

or

(66) 2g(hl(X,Y),N) = ~(LNg)(X,Y). Also

(67) g(hs(X,Y),W) = -g(Y,VxW)

= -g(Y,[X,W])-g(Y,V WX) = g(Y, [W, X]) - Wg(Y, X) + g(VwY, X) = -Wg(X, Y) + g(Y, [W, X}) + g([W, Y],X)+ g(X, VYW) = -(Lwg)(X,Y)-g(W,VYX),

or

(68) 2g(h s (X, Y),W) = ~(Lwg)(X, Y).

Thus from (66) and (68) we have the following theorem. T H E O R E M . Let (M, g, S(TM), 5 ( T M 1 ) ) be a lightlike submanifold of semi-Riemannian manifold (M,g). Then, M is totally geodesic if and only if ltr(TM) and 5 ( T M ± ) are killing subbundles. C O R O L L A R Y . Let (M,g,S(TM),S(TM±)) be a lightlike submanifold of semi-Riemannian manifold (M,g). Then,

g(h\X,Y),N) = g(Y,ANX), g(hs(X,Y),W) = g(Y,AwX).

C O N C L U D I N G R E M A R K S . If screen distribution is integrable then lightlike hypersurface is locally a product C x M', where C is a null curves and M' is a leaf of S(TM), [11]. Particularly, if M is a lightlike surface of a 3-dimensional Lorentz manifold (M ,g) then the integrability of S(TM) implies that M is locally a product C\ x C2, where C\ and C2 are null and spacelike curves of M respectively.

References

[1] A. Bejancu, K. L. Duggal, Lightlike submanifolds of semi-Riemannian manifolds, Acta. Appi. Math. 38 (1995), 197-215.

[2] B. O'Neill, Semi-Riemannian geometry with application to relativity, Academic Press, New York, (1983).

[3] D. N. Kupeli, On null submanifolds in spacetimes, Geom. Dedicata 23 (1987), 33-51. [4] D. N. Kupeli, Degenerate manifolds, Geom. Dedicata 23 (1987), 259-290.


[5] D. N. Kupeli, Degenerate submanifolds in semi-Riemannian geometry, Geom. Dedi-cata 24 (1987), 337-361.

[6] D. N. Kupeli, Singular Semi-Riemannian Geometry, Kluwer, Dordrecht, 1996. [7] E. Kilic, B. Sahin, H. Bayram Karadag, R. Giines, Coisotropic submanifolds of a

semi-Riemannian manifold, Turkish J. Math. 28 (2004), 335-352. [8] K. Honda, Some lightlike submanifolds, SUT J. Math. 37 (2001), 69-78. [9] K. L. Duggal, D. H. Jin, Totally umbilical lightlike submanifolds, Kodai Math. J. 26

(2003), 49-68. [10] K. L. Duggal, A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and

Applications, Kluwer Acad. Publishers, Dordrecht, 1996. [11] K. Yano, Integral Formulas in Riemannian Geometry, Marcel Dekker, New York,

(1970).

DEPARTMENT OF MATHEMATICS UNIVERSITY COLLEGE OF ENGINEERING PUNJABI UNIVERSITY PATIALA 147002, INDIA E-mail: [email protected]

[email protected] [email protected]

Received April 6, 2009; revised version August 4, 2009.

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