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**Stere IANUS¸ ^{*1} and Gabriel Eduard VILCU^{2}**

^{1}Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei Nr.14, sector 1, 70109 Bucure¸sti, Romania **E-mail:** [email protected]

^{2}Department of Mathematics and Computer Science, Petroleum-Gas University of Ploie¸sti, Bulevardul Bucure¸sti Nr. 39, 100680 Ploie¸sti, Romania **E-mail:** [email protected]

- *Corresponding Author:
- Stere IANUS

Faculty of Mathematics and Computer Science,

University of Bucharest, Str. Academiei Nr.14,

sector 1, 70109 Bucure¸sti, Romania

**E-mail:**[email protected]

**Received date: ** December 10, 2007 **Accepted Date: **April 02, 2008

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

In this paper we obtain some properties for real lightlike hypersurfaces of paraquaternionic space forms and give an example.

It is well-known that the lightlike hypersurfaces are of interest in mathematical physics, owing to their extensive use in general relativity and electromagnetism. The general theory of lightlike submanifolds has been developed by Kupelli [4] and Bejancu-Duggal [1]. In [3], the present authors and R. Mazzocco have begun the study of the real lightlike hypersurfaces of paraquaternionic manifolds. In this paper we obtain new properties for this kind of hypersurfaces and give some obstructions to the existence of lightlike hypersurfaces in paraquaternionic space forms.

Let be a (m+2)-dimensional semi-Riemannian manifold with index q ∈ {1, 2, . . . ,m+1} and let (*M, g*) be a hypersurface of , with g = . We say [1] thatM is a lightlike hypersurface of if *g* is of constant rank *m*.

If we consider the vector bundle TM^{⊥} whose fibres are:

then we can easily see that M is lightlike if and only if TM^{⊥} is a distribution of rank 1 on M.

Let *S(TM)* be the complementary distribution of TM^{⊥} in TM, which is called a screen distribution. This distribution is non-degenerate [1]. Thus we have direct orthogonal sum decomposition TM = *S(TM)* ⊥ TM^{⊥}. On the other hand, if *S(TM)*? is the orthogonal complementary vector bundle to *S(TM)* in ,we have direct orthogonal sum = *S(TM)* ⊥ *S(TM)*?.

From [1] we know that, if (*M, g*, *S(TM)*) is a lightlike hypersurface of M, then there exists a unique vector bundle *ltr(TM)* of rank 1 over M so that for any non-zero section of TM^{⊥} on a coordinate neighborhood U ⊂ M, there exists a unique section *N of ltr(TM)* on

The vector bundle * ltr(TM)* is called the null transversal vector bundle of

Let (*M, g*, *S(TM)*) be a lightlike hypersurface of . If is the Levi-Civita connection on , then we have

∇ is a symmetric linear connection on *M* called an induced linear connection, ∇⊥ is a linear connection on the null transversal bundle *ltr(TM)*, h is a Γ(*ltr(TM)*)-valued symmetric bilinear form and *AV* is the shape operator of *M* with respect to *V* . If *{ε,N}* is a pair of sections on U ⊂ M as above, then we define a symmetric *F(U)*-bilinear form B and a 1-form τ on U by

Thus, from (2.1) and (2.2), locally we have

If we denote by P the projection of TM on *S(TM)*, we have the decompositions:

where and ε is a 1-form on *U*. Moreover, it is easy to see that ε = −τ .

Let be a smooth manifold. We say that a rank-3 subbundle σ of is an almost paraquaternionic structure on if a local basis *{J _{1}, J_{2}, J_{3} }* exists on sections of σ, such that for all α ∈ {1, 2, 3} we have

where ε_{1} = 1, ε_{2} = ε_{3} = −1.

Let be a semi-Riemannian manifold and σ an almost paraquaternionic structure on M. The metric is said to be adapted to the paraquaternionic structure σ if it satisfies for all α ∈ {1, 2, 3} and for all vector fields *X, Y* on *M* and any local basis *{J _{1}, J_{2}, J_{3} }* of σ, where ε

for any vector field X on , !1, !2, !3 being local 1-forms over the open for which *{J _{1}, J_{2}, J_{3} }* is a local basis of σ and ε

Let be an almost hermitian paraquaternionic manifold and let (*M, g*) be a real lightlike hypersurface of . We remark that we can choose the screen distribution *S(TM)* such that it contains *J _{α}(TM)^{⊥}* as a vector subbundle, because we have: are tangent to

If is a pair of sections on U ⊂ M as in above section, then and we deduce that J_{α}N 2 Γ(*S(TM)*). It is easy to see that is a vector subbundle of *S(TM)* of rank 6. Then there exists non-degenerate distribution D_{0} on *M* such that

We remark that the distribution D_{0} is invariant with respect to J_{α} for all α 2 {1, 2, 3} (see [3]). We consider now the local lightlike vector fields: for all α 2 {1, 2, 3}. Then any local vector field on *M* is eX_{p}ressed as follows:

where *S* is the projection on the almost paraquaternionic distribution and f_{α} are 1-forms locally defined on *M* by: *f _{α}(X) = g(X,Uα), for all α 2 {1, 2, 3}*.

By using (3.2) we derive:

where Φ_{α}X are the tangential components of J_{α}X for all α 2 {1, 2, 3}.

**Theorem 3.1.** *Let be a paraquaternionic K¨ahler manifold and let M be a lightlike hypersurface of M. Then M is totally geodesic if and only if: for all *

**Proof.** By using (2.3), (2.4) and (3.1) we obtain for any X 2 Γ(TM) and Z_{0} ∈ Γ(D_{0}):

The proof is now complete, since by the definition of a lightlike hypersurface, *M* is totally geodesic if and only if: for all

**Theorem 3.2.*** Let be a paraquaternionic K¨ahler manifold and let M be a lightlike hypersurface of . Then D _{0} is integrable if and only if: C(X, Y ) = C(Y,X), C(X, J_{α}Y ) = C(Y, J_{α}X), B(X, J_{α}Y ) = B(Y, J_{α}X) for all X, Y 2 Γ(D_{0}) and α 2 {1, 2, 3}.*

**Proof. **D_{0} is integrable if and only if: 8X, Y 2 Γ(D_{0}) ) [X, Y ] 2 Γ(D_{0}) and from the general decomposition ? D_{0} ? TM^{⊥}} we deduce that D_{0} is integrable if and only if: for all X, Y 2 Γ(D_{0}), N 2 Γ(*ltr(TM)*) and α 2 {1, 2, 3}. For X, Y 2 Γ(D_{0}) and N 2 Γ(*ltr(TM)*), by using (2.5) we deduce:

From (2.5) and (3.1), we obtain:

and similarly we deduce

The proof is now complete from (3.3), (3.4) and (3.5).

**Corollary 3.1.** *Let be a paraquaternionic K¨ahler manifold and let M be a totally geodesic lightlike hypersurface of M. Then the distribution D is parallel.*

Let be a paraquaternionic K¨ahler manifold and let X be a non-lightlike vector on . Then the 4-plane spanned by {X, J_{1}X, J_{2}X, J_{3} X}, denoted by PQ(X), is called a paraquaternionic 4-plane. Any 2-plane in PQ(X) is called a paraquaternionic plane. A paraquaternionic K¨ahler manifold is said to be a paraquaternionic space form if its paraquaternionic sectional curvatures are equal to a constant.

We recall now the following technical results [3].

**Lemma 4.1.** *Let be a paraquaternionic K¨ahler manifold and let M be a lightlike hypersurface of . Then we have for all α ∈ {1, 2, 3}:*

*(indices mod 3) for all X, Y ∈ Γ(TM).*

Let (*M, g*) be a lightlike hypersurface of a semi-Riemannian manifold . The null sectional curvature of *M at p ∈ M *with respect to , is the real number (M) defined by:

where X_{p} is non-null vector in TpM.

**Theorem 4.1.** There are no real lightlike hypersurfaces with positively or negatively null sectional curvature of paraquaternionic space forms.

**Proof.** Let be a paraquaternionic space form and let *M* be a real lightlike hypersurface of . By using (4.2) and (4.3) we obtain:

If we take now X 2 Γ(D_{0}) we deduce that K»(M) = 0, which proves our assertion.

**Theorem 4.2.** *There are no real lightlike hypersurfaces with positively or negatively Ricci curvature of paraquaternionic space forms.*

**Proof.** Le *M* be a real lightlike hypersurface of a paraquaternionic space form . We have:

where is a basis of *S(TM)*|U. By straightforward computations using (2.3), (2.4), (2.5), (2.6), (4.1) and (4.2) in (4.4) we obtain Ric (X, ) = 0.

We consider the paraquaternionic manifold where the metric g and the structures *J _{1}, J_{2}, J_{3}* are given by

We define now a hypersurface M of by

Thus the tangent space TM is spanned by where

If *X = (x _{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}) ∈ TM^{⊥},* from conditions

*x _{1} = x_{3} = x_{4} = x_{5} = x_{6} = x_{7} = 0, x_{2} = x_{8}*

Hence,

*X = (0, x _{2}, 0, 0, 0, 0, 0, x_{2}) = x_{2}W_{2}*

Thus TM^{⊥} = Sp(W_{2}) and *M* is lightlike hypersurface of

Research was supported by grant D11-22 CEEX 2006-2008.

- BejancuA,DuggalKL (1996) Lightlike submanifolds of semi-Riemannian manifolds and its applications.Kluwer, Dortrecht.
- Garcia-RioE, MatsushitaY, Vasquez-LorenzoR (2001)Paraquaternionic K ¨ahler manifolds. Rocky Mountain J Math 31: 237–260.
- Ianu ¸sS, MazzoccoR, VˆilcuGE(2006) Real lightlikehypersurfaces of paraquaternionic K ¨ahlermanifolds. Mediterr J Math 3: 581–592.
- 4.KupelliN (1996) Singular semi-Riemann geometry.Kluwer, Dortrecht.
- MarchiafavaS (2008) Submanifolds of (para)-quaternionic K ¨ahler manifolds. Note Mat.
- S ¸ahinB,Gune ¸s R (2002)Lightlike real hypersurfaces of indefinite quaternion K ¨ahler manifolds. J Geom75: 151-163

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