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In this paper, we consider Osserman conditions on lightlike warped product (sub-)manifolds with respect to the Jacobi Operator. We define the Jacobi operator for lightlike warped product manifold and introduce a study of lightlike warped product Osserman manifolds. For the coisotropic case with totally degenerates first factor, we prove that this class consists of Einstein and locally Osserman lightlike warped product.

The Riemann curvature tensor is one of the central concepts in the mathematical field of differential geometry. It assigns a tensor to each point of a (semi-)Riemannian manifold that measures the extent to which the metric tensor is not locally isometric to that of Euclidean space. It expresses the curvature of (semi-)Riemannian. Curvature tensor is a central mathematical tool in the theory of general relativity and gravity.

The geometry of a pseudo-Riemannian manifold ( M , g ) is the study of the curvature R ∈ ⊗ 4 T * M which is defined by the Levi-Civita connection ∇ . 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 for example associate to R an endomorphism on tangent bundle of a manifold. In [

On lightlike geometry of hypersurfaces, C. Atindogbe and K. L. Duggal have studied Pseudo-Jacobi operators and considered Osserman conditions [

Let ( M , g ) be a semi-Riemannian manifold ( M , g ) and p ∈ M . An element R ∈ ⊗ 4 T p * M is said to be an algebraic curvature tensor on T p M if R has the following symmetries:

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 ∀ X , Y , Z , W ∈ T p M . (2)

The Riemannian curvature tensor of a Levi-Civita connection is algebraic on T p M for all p ∈ M . If R is an algebraic curvature tensor on T p M , the associated Jacobi operator J R ( X ) with respect to X ∈ T p M is the self-adjoint linear map on T p M characterized by the identity

g ( J R ( X ) Y , Z ) = R ( Y , X , X , Z ) ∀ Y , Z ∈ T p M . (3)

It is obvious that ∀ c ∈ ℝ ⋆ , J R ( c X ) = c 2 J R ( X ) and the domain of J R ( X ) is the unit pseudo-sphere of unit timelike or unit spacelike vectors

S ± ( M ) : = { X ∈ T p M : g ( X , X ) = ± 1 } .

Due to the algebraic properties (1) and (2) of the curvature, we have J R ( X ) X = 0 and g ( J R ( X ) Y , X ) = 0 . Then, the Jacobi operator naturally reduces to the endomorphism J R ( X ) : X ⊥ → X ⊥ .

The Riemannian curvature tensor R of a semi-Riemannian manifold ( M , g ) is said to be a spacelike (resp. timelike) Osserman tensor on T p M if the spectrum spec ( J R ) is constant on S p + ( M ) (resp. S p − ( M ) ). If this is the case at each p ∈ M , we say that ( M , g ) is pointwise Osserman semi-Riemannian manifold.

Motivated by the recent works on lightlike geometry, we consider in this paper lightlike warped product (sub-)manifolds and examine Osserman conditions depending on geometric properties of the factors.

In Section 2, we present background materials of lightlike geometry. In Section 3 we define lightlike warped product Osserman (definition 3.2) and present some important results of our research (Theorem 2, Theorem 3, Theorem 4). Section 4 is concerned with an example given in the neutral semi-Riemannian space R 3 6 ..

Let ( M ¯ , g ¯ ) be a ( m + k ) -dimensional semi-Riemannian manifold of constant index q such that 1 ≤ q < m + k and ( M , g ) be an m-dimensional submanifold of M ¯ . We assume that both m and k are ≥ 1 . At each point p ∈ M ,

T p M ⊥ = { X ∈ T p M ¯ , g ¯ p ( X , Y ) = 0 ∀ Y ∈ T p M } (4)

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 ⊥ (5)

is a smooth distribution with constant rank r > 0 , M is said to be lightlike (or null) submanifold of M ¯ , with nullity 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 ) (6)

where ⊕ 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 ) ⊥ . (7)

A null submanifold M with nullity 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

• r-lightlike if r < min ( m , k ) ;

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

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

• totally null 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 T M in T M ¯ | M , the following relations hold

t r ( T M ) = l t r ( T M ) ⊕ O r t h S ( T M ⊥ ) , (8)

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 ⊥ ) . (9)

The Gauss and Weingarten formulas are

∇ ¯ X Y = ∇ X Y + h ( X , Y ) , (10)

∇ ¯ X V = − A V X + ∇ X t V , (11)

∀ 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 (8), 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 , h s = S ∘ h where ∘ is the composition law, D X l V = L ( ∇ X t V ) , D X s V = S ( ∇ X t V ) . The transformations D l and D s do not define linear connections but Otsuki connections on t r ( T M ) with respect to the vector bundle morphisms L and S. Then,

∇ ¯ X Y = ∇ X Y + h l ( X , Y ) + h s ( X , Y ) (12)

∇ ¯ X N = − A N X + D X l N + D s ( X , N ) (13)

∇ ¯ X W = − A W X + ∇ X s W + D l ( X , W ) (14)

∀ X , Y ∈ Γ ( T M ) , N ∈ Γ ( l t r ( M ) ) and W ∈ Γ ( S ( T M ⊥ ) ) .

Since ∇ ¯ is a metric connection, using (12)-(14) we have

g ¯ ( h s ( X , Y ) , W ) + g ¯ ( Y , D l ( X , W ) ) = g ( A W X , Y ) (15)

g ¯ ( D s ( X , N ) , W ) = g ¯ ( N , A W X ) . (16)

Let P the projection morphism of TM onto S ( T M ) . Using the decomposition (6) we get

∇ X Y = ∇ X * P Y + h * ( X , P Y ) (17)

∇ X ξ = − A ξ * X + ∇ X * t ξ (18)

∀ X , Y ∈ Γ ( T M ) , ξ ∈ Γ ( R a d ( T M ) ) and ∇ * is a metric connection on S ( T M ) .

It follows from (17) and (18) that

g ¯ ( h l ( X , P Y ) ) = g ( A ξ * X , P Y ) (19)

g ¯ ( h * ( X , P Y ) , N ) = g ( A N X , P Y ) (20)

g ¯ ( h l ( X , ξ ) , ξ ) = 0 , A ξ * ξ = 0. (21)

Let R ¯ and R denote the Riemannian curvature tensors on M ¯ and M respectively. The Gauss equation is given by

R ¯ ( X , Y ) Z = R ( X , Y ) Z + A h l ( X , Z ) Y − A h l ( Y , Z ) X + A h s ( X , Z ) Y − A h s ( Y , Z ) X + ( ∇ X h l ) ( Y , Z ) − ( ∇ Y h l ) ( X , Z ) + D l ( X , h s ( Y , Z ) ) − D l ( Y , h s ( X , Z ) ) + ( ∇ X h s ) ( Y , Z ) − ( ∇ Y h s ) ( X , Z ) + D s ( X , h l ( Y , Z ) ) − D s ( Y , h s ( X , Z ) ) (22)

∀ X , Y , Z , U ∈ Γ ( T M ) . Therefore

R ¯ ( X , Y , Z , P U ) = R ( X , Y , Z , P U ) + g ¯ ( h * ( Y , P U ) , h l ( X , Z ) ) − g ¯ ( h * ( X , P U ) , h l ( Y , Z ) ) + g ¯ ( h s ( Y , P U ) , h s ( X , Z ) ) − g ¯ ( h s ( X , P U ) , h s ( Y , Z ) ) . (23)

Definition 2.1 ( [

h ( X , Y ) = g ( X , Y ) H . (24)

Using (10) and (12) it is easy to see that M is totally umbilical if and only if on each coordinate neighbourhood U there exist smooth vector fields H l ∈ Γ ( l t r ( T M ) ) and H s ∈ Γ ( S ( T M ⊥ ) ) such that

h l ( X , Y ) = g ( X , Y ) H l , D l ( X , W ) = 0

h s ( X , Y ) = g ( X , Y ) H s , ∀ X , Y ∈ Γ ( T M ) , W ∈ Γ ( S ( T M ⊥ ) ) . (25)

Definition 2.2 ( [

g ¯ ( h * ( X , P Y ) , N ) = λ g ( X , P Y ) , ∀ X , Y ∈ Γ ( T M | U ) . (26)

Definition 2.3. A coisotropic submanifold ( M , g ) of a semi-Riemannian manifold ( M ¯ , g ¯ ) is screen locally conformal if the local second fundamental forms of the screen distribution S ( T M ) are related with the local second fundamental form of M as follows:

h i * ( X , P Y ) = ϕ i h i l ( X , P Y ) , ∀ X , Y ∈ Γ ( T M ) (27)

where ϕ i is a conformal smooth function on a coordinate neighbourhood U in M. In particular, we say that M is screen homothetic if ϕ i is a non-zero constant.

Let ( M m , g ) be a null submanifold with nullity degree r of a semi-Riemannian manifold ( M ¯ m + k g ¯ ) , ( ξ i ) i and ( N i ) i local frames of Γ ( R a d ( T M ) ) and Γ ( l t r ( T M ) ) respectively satisfying g ¯ ( ξ i , N i ) = δ i j . Consider the 1-forms η i , i = 1, ⋯ , r metrically equivalent to the N i i.e. η i ( . ) = g ¯ ( N i , ⋅ ) . Then, each tangent vector field X has the splitting,

X = P X + ∑ i = 1 r η i ( X ) ξ i . (28)

From now on, we assume that the frames ( ξ i ) i and ( N i ) i are globally defined on M. Consider the Γ ( T * M ) values mapping ♭ g defined on Γ ( T M ) by

X ♭ g : = ♭ g ( X ) = i X g + ∑ i = 1 r η i ( X ) η i (29)

where i X denotes the interior product with respect to X. The mapping ♭ g is an isomorphisme of Γ ( T M ) onto Γ ( T * M ) and we let ♭ g denote its reverse mapping.

i.e.

It is straightforward to check that

Definition 2.4. Let

where

• If

• If

• If all

•

Proposition 1. [

1)

2)

3)

4)

Corollary 1. The leaves

As it is well known, Jacobi operators are associated to algebraic curvature maps (tensors). But contrary to non-lightlike manifolds, the induced Riemann curvature tensor of a lightlike submanifold

In semi-Riemannian case, the relation (3) characterizes the Jacobi operator

that is

For degenerate warped product setting, we consider the associated non-degenerate metric

Definition 3.1. Let

or equivalentently

Definition 3.2. A lightlike warped product submanifold

Theorem 2. Let

Proof. Let

to the conformal class of

product submanifold. Since the Weyl tensor is invariant in the conformal class of a metric, we conclude that

From definition 2.2, it is obvious that if a screen distribution

Due to Proposition 2 in [

Theorem 3. Let

Proof. From (22), the induced Riemannian curvature tensor is

Using (20) and (27), for all

Let

on

Thus the induced Ricci curvature tensor is symmetric and N is locally Einstein. Consider

The pseudo-Jacobi operator

and its characteristic polynomial is

Therefore N is pointwise Osserman. ■

From Proposition 2, theorem 5 in [

Theorem 4. Let

Let

where

Then

Since rank

and we get

It is obvious that M is a coisotropic warped product submanifold of

From (38), we have

and we conclude that

From (39)

and we conclude that the screen distribution

M being coisotropic, taking into account

Let’s consider

The pseudo-Jacobi operator

that is independent of

Remark. From (44), it is obvious that for a lightlike warped product manifold, to be spacelike Osserman or timelike Osserman are equivalent.

Osserman conditions on lightlike warped product manifolds have been considered in this paper. The case of lightlike warped product with the first factor totally degenerate has been explored. Especially in coisotropic case, we have proved that this class consists of Einstein and locally Osserman lightlike warped product. In perspective, we are going to extend this study to other classes of lightlike warped product in order to get later a certain characterization of lightlike warped product Osserman manifolds.

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

Ndayirukiye, D., Nibirantiza, A., Nibaruta, G. and Karimumuryango, M. (2020) Osserman Conditions in Lightlike Warped Product Geometry. Journal of Applied Mathematics and Physics, 8, 585-596. https://doi.org/10.4236/jamp.2020.84045