^{1}

^{*}

^{1}

^{2}

^{3}

Lightlike warped product manifolds are considered in this paper. The geometry of lightlike submanifolds is difficult to study since the normal vector bundle intersects with the tangent bundle. Due to the degenerate metric, the induced connection is not metric and it follows that the Riemannian curvature tensor is not algebraic. In this situation, some basic techniques of calulus are not useable. In this paper, we consider lightlike warped product as submanifold of semi-Riemannian manifold and establish some remarkable geometric properties from which we establish some conditions on the algebraicity of the induced Riemannian curvature tensor.

Semi-Riemannian geometry is the study of smooth manifolds with non-degenerate metric signature [

In 1969, Bishop and O’Neill [

It is well-known that for any semi-Riemannian (warped product) manifold, there is a natural existence for lightlike subspaces. Thus there exists a particular case of submanifolds namely lightlike (degenerate) [

A Riemannian curvature tensor of a semi-Riemannian manifold ( M , g ) is algebraic if it has the following symetry properties

R ( X , Y , Z , W ) = R ( Z , W , X , Y ) = − R ( Y , X , Z , W ) (1)

R ( X , Y , Z , W ) + R ( Y , Z , X , W ) + R ( Z , X , Y , W ) = 0 (2)

∀ X , Y , Z , W ∈ T p M .

The notion of curvature is one of the central concepts of differential geometry, one could argue that is the one central on, distinguishing the geometrical core of the subject from those aspects that are analytic, algebraic, or topological [

Since the whole curvature tensor is difficult to handle, the investigation usually focuses on different objects whose properties allow us to recover curvature tensor. One can associate to R an endomorphism on tangent bundle of a manifold [

Although the lightlike geometry is difficult to study, there are important applications in Physic. In [

In this paper, we examine some conditions on lightlike warped product (sub-)manifolds to admit an algebraic curvature tensor. We particularly consider single lightlike warped product (sub-)manifolds and present some technical and characterization results (Proposition 2, Proposition 3, Proposition 4). We establish algebraicity condition for the (induced) Riemannian curvature tensor on lightlike warped product submanifold (Theorm 5, Theorem 6).

For more details see [

T p M ⊥ = { X ∈ T p M ¯ , g ¯ p ( X , Y ) = 0 , ∀ Y ∈ T p M } (3)

is the normal space at p. In case g ¯ p is non-degenerate on T p M , both T p M and T p M ⊥ are non-degenerate and we have T p M ∩ T p M ⊥ = { 0 } . If the mapping

R a d ( T M ) : p ∈ M ↦ R a d ( T p M ) = T p M ∩ T p M ⊥ (4)

is a smooth distribution with constant rank r > 0 , M is said to be lightlike (or lightlike) submanifold of M ¯ , with lightlikeity degree r. This mapping is called the radical distribution on M. Any complementary (and hence orthogonal) distribution of R a d ( T M ) in TM is called a screen distribution. For a fixed screen distribution on M, the tangent bundle splits as

T M = R a d ( T M ) ⊕ o r t h S ( T M ) . (5)

⊕ o r t h is the orthogonal direct sum. A screen transversal vector bundle S ( T M ⊥ ) on M is any (semi-Riemannian) complementary vector bundle of R a d ( T M ) in T M ⊥ . It is obvious that both S ( T M ⊥ ) and S ( T M ) ⊥ is non-degenerate with respect to g ¯ and

S ( T M ⊥ ) ⊂ S ( T M ) ⊥ . (6)

A lightlike submanifold M with lightlikeity degree r equipped with a screen distribution S ( T M ) and a screen transversal vector bundle S ( T M ⊥ ) is denoted ( M , S ( T M ) , S ( T M ⊥ ) ) . It is said to be

1) r-lightlike if r < min ( m , k ) ;

2) Coisotropic if r = k < m (hence S ( T M ⊥ ) = { 0 } );

3) Isotropic if r = m < k , (hence S ( T M ) = { 0 } );

4) Totally lightlike if r = m = k , (hence S ( T M ) = { 0 } = S ( T M ⊥ ) ).

For any local frame { ξ i } of R a d ( T M ) , there exists a local frame { N i } of sections with values in the orthogonal complement of S ( T M ⊥ ) in S ( T M ) ⊥ such that

g ( ξ i , N j ) = δ i j , g ( N i , N j ) = 0 ,

and it follows that there exists a lightlike transversal vector bundle l t r ( T M ) locally spanned by { N i } .

If we denote by t r ( T M ) a (not orthogonal) complementary vector bundle to TM in T M ¯ | M , the following relations hold

t r ( T M ) = l t r ( T M ) ⊕ o r t h S ( T M ⊥ ) , (7)

T M ¯ | M = T M ⊕ t r ( T M ) = S ( T M ) ⊕ o r t h ( R a d ( T M ) ⊕ l t r ( T M ) ) ⊕ o r t h S ( T M ⊥ ) . (8)

The Gauss and Weingarten formulas are

∇ ¯ X Y = ∇ X Y + h ( X , Y ) , (9)

∇ ¯ X V = − A V X + ∇ X t V , (10)

∀ X , Y ∈ Γ ( T M ) , V ∈ Γ ( t r ( T M ) ) . The components ∇ X Y and − A V X belong to Γ ( T M ) , h ( X , Y ) and ∇ X t V to Γ ( t r ( T M ) ) . ∇ and ∇ t are linear connections on TM and the vector bundle t r ( T M ) respectively. According to the decomposition (7), let L and S denote the projection morphisms of t r ( T M ) onto l t r ( T M ) and S ( T M ⊥ ) respectively, h l = L ∘ h ,

Since

Let P the projection morphism of TM onto

It follows from (16) and (17) that

Let

Definition 2.1. [

Using (9) and (11) its is easy to see that M is totally umbilical if and only if on each coordinate neighbourhood

Definition 2.2. [

Definition 2.3. A coisotropic submanifold

where

Definition 2.4. Let

where

• If

• If

• If all

•

For a singly warped product, we have the following:

Proposition 1. [

1)

2)

3)

4)

From the previous proposition, one can see that

Definition 2.5. A lightlike warped product submanifold

In the following, we consider a lightlike warped product

Proposition 2. Let f be a coisotropic isometric immersion of a warped product

Proof. In case of coisotropic submanifold we have

Thus

Then

that is

Proposition 3. Any totally umbilical lightlike warped product submanifold of a semi-Riemannian manifold is mixed totally geodesic.

Proof. From the expressions (27) and (2.1), we have

Proposition 4. Let

1)

2)

3)

4)

5)

Proof. Let

From (16) we have

From (19) and (30) we have

Let

Using (11), (16), (29) and (30) we have

Moreover

We give the following result on the algebraic properties of the induced Riemannian tensor on lightlike warped product with the first factor totally degenerate.

Theorem 5. Let

Proof. The result hold from Theorem 3.2 in [

In case of coisotropic warped product of a semi-Riemannian manifold with constant sectional curvature which is conformal screen, we establish the following.

Theorem 6. Let

Proof. Since

It is then obvious that

and we infer

The algebraicity conditions of the induced Riemannian curvature tensor have been explored in this paper. Some remarkable geometric properties of lightlike warped product submanifolds have been given. From the above results, one can see that the induced Riemannian curvature tensor on lightlike warped product submanifolds with totally null first factor is an algebraic curvator tensor. In the future, we will be studying Osserman conditions on lightlike warped product manifolds.

The authors declare no conflicts of interest regarding the publication of this paper.

Ndayirukiye, D., Nibaruta, G., Karimumuryango, M. and Nibirantiza, A. (2019) Algebraicity of Induced Riemannian Curvature Tensor on Lightlike Warped Product Manifolds. Journal of Applied Mathematics and Physics, 7, 3132-3139. https://doi.org/10.4236/jamp.2019.712220