REAL HYPERSURFACES OF INDEFINITE KAEHLER MANIFOLDS

We show the existence of ( ϵ )-almost contact metric structures and give examples of ( ϵ )-Sasakian manifolds. Then we get a classification theorem for real hypersurfaces of indefinite complex space-forms with parallel structure vector field. We prove that ( ϵ )-Sasakian real hypersurfaces of a semi-Euclidean space are either open sets of the pseudosphere S2S2n

The manifold M is supposed to be paracompact and differentiable of class O. Denote F(M) the algebra of real differentiable functions on M and by F(TM) the F(M)-module of differentiable vector fields on M The same notation is used for the set of sections of a vector bundle over M or over any other madfold.
Throughout the paper, by a semi-ILiemannian metric on M we understand a ,ion-degenerate symmetric tensor field g of type (0,2), (cf.O'Neill [8]).We now suppose on M there exists a semi-Pdemannian metric # (see Duggal [5]) that satisfies g(.fX, .fY)g(X, Y) erl(X)l(Y), VX, Yer(TM) (1.3) where e +1 It follows that (X) eg(X,), VXeF(TM) (1.4) and e g(,). (1.,5) Hence is never a light-like vector field on M This implies that the contact distribution D {X F(TM), r/(X)= 0} is always non-degenerate on M Moreover, thc index of g is an odd number v 2r + 1 in case is time-like and an even number v 2r otherwisc.This follows as a consequence of the fact that on M we may consider an orthonormal ficld frame {E,... ,E,, fE,... ,fE,, } with Ei F(D) and such that g(Ei,Ei) g(fEi, fEi).
We are now concerned with the existence of semi-Riemannian metrics satisfying (1.3).In the particular case e 1 and r, 0 there exists a Riemannian metric g satisfying (1.3) and M is the usual almost contact metric manifold (cf.Blair [4]).For the general case, following Blair [4], and subject to the above mentioned restrictions of the index of g we have the following result.THEOREM 1.Let (f,,r/) be an almost contact structure and h0 be a semi-Riemannian metric on M such that is not a light-like vector field.Then there exists on M a symmetric tensor field g of type (0,2) satisfying (1.3) h0 where a h0 (, ) and PROOF.We first define two semi-Riemannian metrics hih(Z,Y) h(.f2X,.f2Y)+ e,(X),(Y),VX, Y e r(TM).
In order to prove that h is a semi-Riemannian metric we first note that rl(X) eh(X, )and h(, ,) e.
Therefore, in general, the above theoren does not provide us a seani-Riexnannian metric on M satisfying (1.3).However, we may prove the existence of Lorentz metrics satisfying (1.3).
COROLLARY 1.Let (f,,r/) be an ahnost contact structure on M. Then there exists a Lorcntz metric g on M satisfying (1.3) with e -1.
PROOF.Since M is paracompact there exists a Riemannian metric ho on M We define b,1, h and g as in Theorem 1 with e -1.Then it is easy to see that both It and g arc Lorcntz metrics on M Besides, g satisfies (1.3) with e -1.
We call (f,,,rl, g satisfying (1.1) and (1.3) an (e)-almost contact metric structure and M an (e)zalmost contact metric manifold.Thus we have the following new classes of maxfifolds.
1 e 1 and u 2r.M is called a space-like almost contact metric manifold.2 e -1 and u 2r + 1. M is cflled a time-like almost contact metric mafifol(t. An important subclass of the second class is the Lorentz almost contact manifold (e -1, u 1),recently studied by the second author (see Duggal [5]).As following the terminology of DuggaJ [5] and the definition of space-time (scc Becm-Ehrlicl [2]) a time orientable Lorentz almost contact manifold will be called a contact space-time, ttcrc for the sake of completeness, we state the following result (proved in Duggal [5]) on contact space-times.THEOREM 2. (Duggal[5]).For an (e)-almost contact metric manifold M, the following are equivalent-(1) M is contact space-time.
(2) The characteristic vector field is time-like and the 2n-dimensional contact distribu- tion (n, jr, g/n) is space-like.Next, we consider the fundamental 2-form of the (e)-almost contact metric structure defined by '(X,Y) g(X, fY),VX, Y F(TM) (1.6) Then we say that (f,,rt, g is an (e)-contact metric structure if we have (X, Y) dr(X, Y), VX, Y F(TM). (1.7) In this case M is an (e)-contact metric manifold.Besides we recall that the almost contact structure (f,,r/)is normal if where [f,f] is the Nijcnhuis tensor field associated to f.An (e)-contact metric structure which is normal is called an (e)-Sasakian structure.A nanifold endowed with an ()-Sasakian structure is called an ()-Sasakian manifold.As in the case of Riemamfian Sasakian manifolds we havc.THEOREM 3.An ()-Mmost contact metric structure (f, C, r/, g) is ()-Sasakian if and only if (Vxf)Y=g(X,Y)-e7(Y)X, VX, Y r(TM) (1.9 where X7 is the Levi-Civita connection with respect to g If wc replace Y by in (1.9) we get Vx =-eIX, VX F(TM).
(1.10) Thus, we have: COROLLARY 2. The characteristic vector field on an (e)-Sasakian manifold is a Killing vector field.
Sasakian manifolds with indefinite metrics have been first considered by Takahashi [9].
Their importance for physics has been pointed out by one of the present authors (see Duggal   According to the causal character of we have two new classes of (e)-Sasakian manifolds.
Thus in case is space-like (e 1 and r, 2r), (resp.time-like, e -1 mad v 2r + 1)   we say that M is a space-like Sasakian manifold (resp.time-like Sasakian manifold}.In case e 1 and v 0 we get the well-known concept of Riemannian Sasakian mafifold.Ccrt,'finly for physics it is important to consider Lorentz metrics.In this case e -1, v 1 and wc call M a Lorentz-Sasakian manifold or a Sasakian-spacetime (cf.Duggal [5]).
We close the section with some examples of (e)-Sasakian structures on R2"+x.Other examples we shall give in section 3. First we make the following notations: Ov, the p x k null matrix I the k x k unit matrix.For any non-negative integer s < n we put -1 for ae {1,... ,s} e }, in case s -0, 1 for a {s+l,. ,n and e 1 in case s 0 yi z) 1 n as cartesian coordinates on R 2"+ and define with Then we consider (z', respect to the natural field of frames { 0 o o} -,, --,, a tensor field f of type (1,1) by its matrix.0,,,, I,, 0,,, ] [f]= --[n On,n 0n,1 01,n eaya 0 The differential 1-form 7 is defined by (1.12) 1=dz-y'dz if s=0. .= The vector field is defined for each s 1)y (1.13) 0 2ezz (1.14) It is easy to check (1.1) and thus (f,.l) is an almost contact structure on R 2''+1 for each s E {0,1 n} Finally, we define the scmi-Rienaannian metric g by thc matrix (1.15) for s # 0, and .] 4-y'y On,, y' 0.,.
PHYSICAL EXAMPLE.First we need the following information (for details see [2,8].Let M be a spacetime manifold, with a Lorentz metric g of signature (-, +,... +).A spacetime M is called globally hyperbolic if M is a product manifold of the form (M R S,g -dr +G) with (S, G) a compact Riemannian mafifold.Recently the second author, Duggal [5], has proved the following physical result, also valid for Sasakian structures.THEOREM 4 (Duggal [5]).An odd dinensional globally hyperbolic spacetimc can carry t Lorcntz-Sasakian structure.
In another direction, physically, Corollary 2 of Theorem 3 is important for the special case of Sasakian spacetines since is a Killing vector field.The existence of Killing vector fields in spacetimes has often been used as the most effective symmetry.In fact, many exact solutions of Einstein field equations have been found by assuming one or more Killing vector fields (Kramer-Stephani-Herlt [6]).

REAL HYPERSURFACES OF INDEFINITE KAHLER MANIFOLDS.
Let hS/ be a real 2(n + 1)-dimensional manifold.Suppose is endowed with an almost complex structure .]and a semi-Riemannian metric t} satisfying It follows that the index of t is an even number t, 2(r + 1).Then we say that is an indefinite almost Hcrmitian manifold.Moreover, if on we have (gxY)Y o, for any X,Y r(T.), (

2.2)
where 7 is the Lcvi-Civita connection with respect to ., we say that is aax indefinite Kalderian manifold (see narros-Romero [1]).Now suppose M is an orientable non-degenerate real hypersurface of// Let N be the normal unit vector field of M Thus by (2.1) and taking account of the orientability of M wc see that -3VN is a vector field tangent to M. Then the equations of Gauss and Weingartcn are given by xY VxY + h(X,Y)N, VX, Y e r(TM), (2.3) and xN =-AX, VX r(TM), (2.4) respectively, where X7 is the Levi-Civita connection with respect to the scmi-Rienannim metric g induced by on M A is the shape operator of M and h is a symmetric tensor field of type (0,2) on M Suppose now [l(g,g) e and by (2.1) we have g(,) e. Whc'n from (2.3) and (2.4) we get h(X, Y) eg(AX, Y), VX, Y e r(TM).Hence (2.3) becomes TxY 7xY + eg(AX, Y)N, VX, Y C r(TM). (2.5) We now denote by {}the distribution spanned by on M and by D the complementary orthogonal distribution to {} in TM.Certainly D is invariant by a and the distribution {} is carried by a into the normal bundle.Thus any real hypersurface of an indefinite Kahler manifold is an example of a CR-submanifold (see Bejancu [31).The projection morphism of TM to D is then denoted by P Hence any vector field X on M is written as follows X PX + rl(X), (2.6) where r/is a 1-form on M defined by n(x) (x,o. (2.z) Thus we have r/() 1. (2.8) Further, we define a tensor field f on M by IX ]PX, VX r(TM). (2.9) Then taking account that D is invariant by J we get X -Z + v(Z).
for any X,Y F(TM).
Then we replace Y in (2.12) by and obtain (2.14).
From Proposition 2 we easily obtain .
Therefore M is a totally umbilical hypersurface (but not totally geodesic) with normal curva- ture k -e Hence by Lelnma 35 and Proposition 36 from O'Neill [8], p.l16, wc obtain that M has constant curvature e and it is an open set of '28 (I) when e I and an open set of z,_a (1) when e -1.
Suppose now M is a totally umbilical real hypcrsurface of .,/,that is, A pI, where p is a differentiable function and I is the identity on F(TM).PROOF.The first part of the assertion follows from the proof of Theorem 7. Suppose now M is totally umbilical with p -e. Then A -e and thus r/(A) -,.Hence (3.1) is satisfied and tiffs completes the proof.
REMARK 1. Tashiro [10] has constructed the Sasakian structure on a sphere of a Euclidean space and Takahashi [9], by a different approach than ours, obtained the (e)-Sasakian structure q,2n+l 2n+l H2,_1 (1) on,:s (1) and Now suppose M is a totally umbilical real hypersurface of an indefinite complex space form //(c). Then we get g ((VxA)Y (VyA)X,) O, VX, Y r(D). (3.8) On the other hand, from (2.18) we get c 9((X,Y),N)g(X, fY), VX, Y F(D). (3.9) Hence from (3.8) and (3.9), taking account of (2.17) we.obtain c 0, which enable us to state PROPOSITION 3.There exist no totally umbilical real hypersurfaces in an indefinite complex space form of non-null holomorphic sectional curvature.
Tashiro-Tachibana [11] first obtained such a result for positive definite complex space forms.
Hence we have the assertion 1 of the theorem.The assertion 2 follows from (4.13) taking into account that the cigenvalues of A are supposed to be real.COROLLARY 5. Let M be either a space-like cosymplectic real hypersurface of rm indefi- nite complex space-form of positive holomorphic sectional curvature or a time-like cosymplectic real hypersurface of an indefinite complex space-form of negative holomorphic sectional curvature.Then the shape operator of M has at least two eigenvalues which are not real.
REMARK 3. In the case of cosymplectic real hypersurfaces of positive definite space forms, important results have been obtained by Okumura [7].
Next by (4.1) we see that the distribution D is involutive on an (e)-cosymplectic real hypersurface M Moreover, in case 2 of Theorem 9 by using (4.4) we derive that A has eigenvalues (+1) and (-1) with the same multiplicity n.Denote by D + and D-the eigen distributions with respect to the above eigenvalues.Further, take X, Y q F(D+), Z q F(D) and from (2.17) we get ) we obtain (VxA) Y (VrA)X 0, VX, Y e F(TM) (3.4) Next, from (3.1), we get AX -eX, VX r(D), and (3.3).