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Hypersurfaces of paraquaternionic space forms 1 | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Hypersurfaces of paraquaternionic space forms 1

Stere IANUS¸*1 and Gabriel Eduard VILCU2

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

2Department 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

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Abstract

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

Introduction

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.

Lightlike hypersurfaces

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

If we consider the vector bundle TM whose fibres are:

equation

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 equation,we have direct orthogonal sum equation = 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 equation of TM on a coordinate neighborhood U ⊂ M, there exists a unique section N of ltr(TM) on U satisfying:

equation

The vector bundle ltr(TM) is called the null transversal vector bundle of M with respect to S(TM). Thus, we have

equation

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

equation

equation

∇ 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

equation

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

equation

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

equation

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

Real lightlike hypersurfaces of paraquaternionic manifolds

Let equation be a smooth manifold. We say that a rank-3 subbundle σ of equation is an almost paraquaternionic structure on equation if a local basis {J1, J2, J3 } exists on sections of σ, such that for all α ∈ {1, 2, 3} we have

equation

where ε1 = 1, ε2 = ε3 = −1.

Let equation be a semi-Riemannian manifold and σ an almost paraquaternionic structure on M. The metric equation is said to be adapted to the paraquaternionic structure σ if it satisfies equation for all α ∈ {1, 2, 3} and for all vector fields X, Y on M and any local basis {J1, J2, J3 } of σ, where ε1 = 1, ε2 = ε3 = −1. Moreover equation is said to be an almost hermitian paraquaternionic manifold. Moreover, if the Levi-Civita connection of equation satisfies the following conditions for all α ∈ {1, 2, 3}:

equation

for any vector field X on equation, !1, !2, !3 being local 1-forms over the open for which {J1, J2, J3 } is a local basis of σ and ε1 = 1, ε2 = ε3 = −1, then equation is said to be a paraquaternionic K¨ahler manifold (see [2]).

Let equation be an almost hermitian paraquaternionic manifold and let (M, g) be a real lightlike hypersurface of equation. We remark that we can choose the screen distribution S(TM) such that it contains Jα(TM) as a vector subbundle, because we have: equation equation are tangent to M and thus Jα(TM) is a 3-rank distribution on M such that Jα(TM) \ TM = {0}.

If equation is a pair of sections on U ⊂ M as in above section, then equation and we deduce that JαN 2 Γ(S(TM)). It is easy to see that equation is a vector subbundle of S(TM) of rank 6. Then there exists non-degenerate distribution D0 on M such that equation

We remark that the distribution D0 is invariant with respect to Jα for all α 2 {1, 2, 3} (see [3]). We consider now the local lightlike vector fields: equation for all α 2 {1, 2, 3}. Then any local vector field on M is eXpressed as follows:

equation

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

By using (3.2) we derive:

equation

where ΦαX are the tangential components of JαX for all α 2 {1, 2, 3}.

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

Proof. By using (2.3), (2.4) and (3.1) we obtain for any X 2 Γ(TM) and Z0 ∈ Γ(D0):

equation

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

Theorem 3.2. Let equation be a paraquaternionic K¨ahler manifold and let M be a lightlike hypersurface of equation. Then D0 is integrable if and only if: C(X, Y ) = C(Y,X), C(X, JαY ) = C(Y, JαX), B(X, JαY ) = B(Y, JαX) for all X, Y 2 Γ(D0) and α 2 {1, 2, 3}.

Proof. D0 is integrable if and only if: 8X, Y 2 Γ(D0) ) [X, Y ] 2 Γ(D0) and from the general decomposition equation ? D0 ? TM} we deduce that D0 is integrable if and only if: equation for all X, Y 2 Γ(D0), N 2 Γ(ltr(TM)) and α 2 {1, 2, 3}. For X, Y 2 Γ(D0) and N 2 Γ(ltr(TM)), by using (2.5) we deduce:

equation

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

equation

and similarly we deduce

equation

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

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

Non-existence of real lightlike hypersurfaces in paraquaternionic space form

Let equation be a paraquaternionic K¨ahler manifold and let X be a non-lightlike vector on equation. Then the 4-plane spanned by {X, J1X, J2X, J3 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 equation be a paraquaternionic K¨ahler manifold and let M be a lightlike hypersurface of equation. Then we have for all α ∈ {1, 2, 3}:

equation

(indices mod 3) for all X, Y ∈ Γ(TM).

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

equation

where Xp 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 equation be a paraquaternionic space form and let M be a real lightlike hypersurface of equation. By using (4.2) and (4.3) we obtain:

equation

If we take now X 2 Γ(D0) 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 equation. We have:

equation

where equation 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, equation) = 0.

Example

We consider the paraquaternionic manifold equation where the metric g and the structures J1, J2, J3 are given by

equation

We define now a hypersurface M of equation byequation

Thus the tangent space TM is spanned by equation where

equation

If X = (x1, x2, x3, x4, x5, x6, x7, x8) ∈ TM, from conditions equation(X,Wi) = 0 (i 2 {1, ..., 7}) we obtain

x1 = x3 = x4 = x5 = x6 = x7 = 0, x2 = x8

Hence,

X = (0, x2, 0, 0, 0, 0, 0, x2) = x2W2

Thus TM = Sp(W2) and M is lightlike hypersurface of equation

Acknowledgement

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

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