LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE SASAKIAN MANIFOLD WITH A NON-METRIC θ-CONNECTION

Abstract

In this paper, we study two types of 1-lightlike submanifolds, named by lightlike hypersurface and half lightlike submanifold, of an indefinite Sasakian manifold admitting non-metric ${\theta}$-connections. We prove that there exist no such two types of 1-lightlike submanifolds of an indefinite Sasakian manifold.

1. INTRODUCTION

A linear(affine) connection on a semi-Riemannian manifold (, ) is called a non-metric θ-connection if, for any vector felds X, Y and Z on , it satisfes

where θ is a 1-form, associated with a non-vanishing smooth vector field ζ by

Two special cases are important for both the mathematical study and the applications to physics: (1) A non-metric θ-connection on is called a semi-symmetric non-metric connection if its torsion tensor satisfes

The notion of semi-symmetric non-metric connections on a Riemannian manifold was introduced by Ageshe and Chafle [1] and later studied by many authors. The lightlike version of Riemannian manifolds with semi-symmetric non-metric connections have been studied by some authors [11, 12, 13, 14, 17].

(2) A non-metric θ-connection is called a quarter-symmetric non-connection if its torsion tensor satisfies

where ϕ is a (1, 1)-type tensor field. The quarter-symmetric non-metric connection was introduced by S. Golad [8], and then, studied by many authors [2, 3, 16].

The theory of lightlike submanifolds is an important topic of research in differential geometry due to its application in mathematical physics. The study of such notion was initiated by Duggal and Bejancu [4] and later studied by many authors [6, 7]. Although now we have lightlike version of a large variety of Riemannian submanifolds, the geometry of lightlike submanifolds of indefinite Sasakian manifolds admitting non-metric θ-connections has not been introduced as yet.

In this paper, we study the geometry of two type of lightlike submanifolds, named by lightlike hypersurface and half lightlike submanifold, of an indefinite Sasakian manifold admitting non-metric θ-connection, in which the 1-form θ and its associated vector field ζ, defined by (1.1), is identical with the structure 1-form θ and its associated vector field ζ, respectively, of the indefinite Sasakian structure (J, ζ, θ, ). We prove the following result:

• There exist no such two types of 1-lightlike submanifolds of an indefinite Sasakian manifold admitting non-metric θ-connections.

From these results we deduce to the following result:

• There exist no such two types of 1-lightlike submanifolds of an indefinite Sasakian manifold admitting either semi-symmetric non-metric connection or quarter-symmetric non-metric connection.

 

2. NON-EXISTENCE THEOREM FOR LIGHTLIKE HYPERSURFACES

An odd-dimensional semi-Riemannian manifold (, ) is said to be an indefinite Sasakian manifold ([9]~[10]) if there exists a structure set {J, ζ, θ, }, where J is a (1, 1)-type tensor field, ζ is a vector field which is called the structure vector field of and θ is a 1-form such that, for any vector fields X and Y on ,

holds, where ∊ = 1 or ‒1 according as ζ is spacelike or timelike respectively.

In this case, we show that Jζ = 0 and θ ○ J = 0. The structure set {J, ζ, θ, } is called an indefinite Sasakian structure of . From (2.1) and (2.2), we get

In the entire discussion of this article, we shall assume that the structure vector field ζ of to be unit spacelike, i.e., ∊ = 1, without loss of generality.

Let (M, g) be a lightlike hypersurface, with a screen distribution S(TM), of an indefinite Sasakian manifold (, ). We follow Duggal and Bejancu [4] for notations and structure equations used in this section. For any null section ξ of TM⊥ on a coordinate neighborhood 𝒰 ⊂ M, there exists a unique null section N of a unique vector bundle tr(TM) in S(TM)⊥ satisfying

We call tr(TM) and N the transversal vector bundle and the null transversal vector field of M with respect to the screen distribution respectively. Let P be the projection morphism of TM on S(TM). Then the local Gauss and Weingartan formulas of M and S(TM) are given respectively by

for any X, Y ∈ Γ(TM), where the symbols ∇ and ∇* are the induced linear connections on TM and S(TM) respectively, B and C are the local second fundamental forms on TM and S(TM) respectively, AN and are the shape operators on TM and S(TM) respectively and τ and σ are 1-forms on TM.

The induced connection ∇ on M is not metric and satisfies

for any X, Y, Z ∈ Γ(TM), where η is a 1-form on TM such that

From the fact that , we know that B is independent of the choice of the screen distribution S(TM), and satisfies

From this result and (2.4), for all X ∈ Γ(TM) we obtain

Now we set a = θ(N) and b = θ(ξ). For all X, Y ∈ Γ(TM), the above second fundamental forms B and C are related to their shape operators by

Now we quote the following result by Jin [9]:

Lemma 1. Let M be a lightlike hypersurface of an indefinite almost contact metric manifold . Then J(TM⊥) and J(tr(TM)) are subbundles of S(TM), of rank 1.

Theorem 2.1. There exist no lightlike hypersurfaces of an indefinite Sasakian manifold admitting a non-metric θ-connection.

Proof. Now we consider two vector fields V and U on S(TM) such that

For any X ∈ Γ(TM), the action JX of X by J is expressed as

where FX is the tangential component of JX and u is a 1-form given by

Applying X to (2.13)1 and using (2.2), (2.4), (2.10) and (2.14), we have

On the other hand, taking Y = V to (2.11) and using (2.15), we have

From the last two equations, we obtain bu(X) = 0 for all X ∈ Γ(TM). Taking X = V to this result and using (2.1), we get b = 0. This implies that ζ is tangent to M. Replacing Y by ζ to (2.4) and using (2.3), we obtain

Applying ∇X to g(ζ, ζ) = 1 and using (2.8), we get

Substituting (2.16)1 into the last equation and using (2.15) and (2.16)2, we have θ = 0 on TM. It is a contradiction as θ(ζ) = 1. Thus there exist no lightlike hypersurfaces of an indefinite Sasakian manifold admitting a non-metric θ-connection.

Corollary 1. There exist no lightlike hypersurfaces of an indefinite Sasakian manifold admitting a semi-symmetric non-metric connection or a quarter-symmetric non-metric connection.

 

3. NON-EXISTENCE THEOREM FOR HALF LIGHTLIKE SUBMANIFOLDS

Let (M, g) be a half lightlike submanifold, with a screen distribution S(TM) and the radical distribution Rad(TM), of an indefinite Sasakian manifold (, ). We follow Duggal and Jin [5] for notations and structure equations used in this section. For any null section ξ of Rad(TM), there exists a uniquely defined lightlike vector bundle ltr(TM) and a null vector field N of ltr(TM) satisfying

We call N, ltr(TM) and tr(TM) = S(TM⊥)⊕orth ltr(TM) the lightlike transversal vector field, lightlike transversal vector bundle and transversal vector bundle of M with respect to S(TM) respectively. In this case, the local Gauss and Weingartan formulas of M and S(TM) are given by

for all X, Y ∈ Γ(TM), where ∇ and ∇* are induced linear connections on TM and S(TM) respectively, B and D are called the local second fundamental forms of M, C is called the local second fundamental form on S(TM). AN, and AL are linear operators on TM and τ, ρ, ϕ and σ are 1-forms on TM.

Using (1.1) and (3.1), for all X, Y, Z ∈ Γ(TM) we have

From the facts and , we know that B and D are independent of the choice of S(TM) and satisfy

From this result and (3.1), for all X ∈ Γ(TM) we obtain

We set b = θ(ξ), a = θ(N) and e = θ(L). For any X, Y ∈ Γ(TM), the above three local second fundamental forms are related to their shape operators by

Now we quote the following result by Jin [10]:

Lemma 2. Let M be a half lightlike submanifold of an indefinite almost contact metric manifold . Then the distributions J(TM⊥), J(tr(TM)) and J(S(TM⊥)) are vector subbundles of S(TM), of rank 1.

Theorem 3.1. There exist no half lightlike submanifolds of an indefinite Sasakian manifold admitting a non-metric θ-connection.

Proof. Now we consider three vector fields V, U and W on S(TM) such that

For any X ∈ Γ(TM), the action JX of X by J is expressed as

where FX is the tangential component of JX and u and w are 1-forms given by

Applying to (3.12)1 and using (2.2), (3.1), (3.8) and (3.13), we have

On the other hand, taking Y = V to (3.9) and using (3.14)1, we have

From this and (3.16)1, we obtain bu(X) = 0 for any X ∈ Γ(TM). Thus we get b = 0. It follow that and τ = σ. Applying to (3.12)3 and using (2.2), (3.1), (3.3), (3.12) and (3.13), we have

On the other hand, taking Y = W to (3.10), we have

From this and (3.18)2, we obtain ew(X) = 0 for any X ∈ Γ(TM). Thus we get e = 0. As b = e = 0, the structure vector field ζ is tangent to M.

Applying to (3.13) and using (3.1) ~ (3.3), (3.12) and (3.13), we have

(∇XF)Y = u(Y)ANX + w(Y)ALX − B(X, Y)U − D(X, Y)W + g(X, Y)ζ − θ(Y)X,

(∇Xu)Y = −u(Y)τ (X) − w(Y)ϕ(X) − B(X, FY),

(∇Xw)(Y) = −u(Y)ρ(X) − D(X, FY).

On the other hand, applying ∇X to u(Y) = g(Y, V) and w(Y) = g(Y,W) by turns and using (3.6), (3.15), (3.16)1, (3.17), (3.18)1 and θ ○ J = 0, we have

(∇Xu)(Y) = − u(Y)τ(X) − w(Y)ϕ(X) − B(X, FY) − θ(Y)u(X) ,

(∇Xw)(Y) = − u(Y)ρ(X) − D(X, FY) − θ(Y)w(X).

From the last four equations, for all X, Y ∈ Γ(TM) we obtain

Taking X = U and Y = ζ to the first equation, or taking X = W and Y = ζ to the second equation, we have 1 = 0. It is a contradiction. Thus there exist no half lightlike submanifolds of an indefinite Sasakian manifold admitting a non-metric θ-connection.

Corollary 2. There exist no half lightlike submanifolds of an indefinite Sasakian manifold admitting a semi-symmetric non-metric connection or a quarter-symmetric non-metric connection.