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Journal of Inequalities and Applications

Erschienen in:

Open Access 01.12.2013 | Research

Chen-like inequalities on lightlike hypersurfaces of a Lorentzian manifold

verfasst von: Mehmet Gülbahar, Erol Kilic̣, Sadık Keleṣ

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2013

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Abstract

We introduce k-Ricci curvature and k-scalar curvature on lightlike hypersurfaces of a Lorentzian manifold. We establish some inequalities between the extrinsic scalar curvature and the intrinsic scalar curvature. Using these inequalities, we obtain some characterizations on lightlike hypersurfaces. We give some results with regard to curvature invariants and

S (n_{1}, \dots, n_{k})

-spaces for lightlike hypersurfaces of a Lorentzian manifold.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors did not provide this information.

1 Introduction

In 1873, Schläfli conjectured that every Riemann manifold could be locally considered as a submanifold of an Euclidean space with sufficient high codimension. This was proven by Janet in [1], Cartan in [2]. Friedmann extended the theorem to semi-Riemannian manifolds in [3]. Chen gave a relation between the sectional curvature and the shape operator for an n-dimensional submanifold M in a Riemannian space form

R^{m} (\bar{c})

in [4] as follows:

A_{N} > \frac{n - 1}{n} (c - \bar{c}) I_{n},

(1.1)

where

A_{N}

is a shape operator of M,

c = inf K \neq \bar{c}

and

I_{n}

is an identity map. Also, Chen established a sharp inequality between the main intrinsic curvatures (the sectional curvature and the scalar curvature) and the main extrinsic curvatures (the squared mean curvature) for a submanifold in a real space form

R^{m} (\bar{c})

in [5] as follows:

For each unit tangent vector

X \in T_{p} M^{n}

H^{2} (p) \geq \frac{4}{n^{2}} {Ric (X) - (n - 1) \bar{c}},

(1.2)

where

H^{2}

is the squared mean curvature and

Ric (X)

is Ricci curvature of

M^{n}

at X.

In [6], Hong and Tripathi presented a general inequality for submanifolds of a Riemannian manifold by using (1.2). In [7], this inequality was named Chen-Ricci inequality by Tripathi.

In [8] and [9], Chen introduced a Riemannian invariant

δ (n_{1}, \dots, n_{k})

δ (n_{1}, \dots, n_{k}) = τ (p) - inf {τ_{π_{1}} (p) + \dots + τ_{π_{k}} (p)},

(1.3)

where

τ (p)

is scalar curvature of M,

τ_{π_{j}} (p)

is j-scalar curvature,

π_{1}, \dots, π_{k}

run over all k mutually orthogonal subspaces of

T_{p} M

such that

dim π_{j} = n_{j}

j = 1, \dots, k

. In [10], the authors gave optimal relationships among invariant

δ (n_{1}, \dots, n_{k})

, the intrinsic curvatures and the extrinsic curvatures.

Later, Chen and some authors found inequalities for submanifolds of different spaces. For example, these inequalities were studied at submanifolds of complex space forms in [11‐13]. Contact versions of Chen inequalities and their applications were introduced in [7, 14‐16]. In [17], Tripathi investigated these inequalities in curvature-like tensors. Furthermore, Haesen presented an optimal inequality for an m-dimensional Lorentzian manifold embedded as a hypersurface on an

(m + 1)

-dimensional Ricci-flat space in [18]. The authors in [19] proved an inequality using the extrinsic and the intrinsic scalar curvature in a semi-Riemannian manifold. In [20], Chen introduced space-like submanifolds (Riemannian submanifolds) of a semi-Riemannian manifold.

As far as we know, there is no paper about Chen-like inequalities and curvature invariants in lightlike geometry. So, we introduce k-plane Ricci curvature and k-plane scalar curvature of a lightlike hypersurface of a Lorentzian manifold. Using these curvatures, we establish some inequalities and by means of these inequalities, we give some characterizations of a lightlike hypersurface on a Lorentzian manifold. Finally, we introduce the curvature invariant

δ (n_{1}, \dots, n_{k})

on lightlike hypersurfaces of a Lorentzian manifold.

2 Preliminaries

Let

(\tilde{M}, \tilde{g})

be an

(n + 2)

-dimensional semi-Riemannian manifold and M be a lightlike hypersurface of

\tilde{M}

. The radical space or the null space of

T_{p} M

, at each point

p \in M

, is a one-dimensional subspace

Rad T_{p} M

defined by

Rad T_{p} M = {ξ \in T_{p} M : g_{p} (ξ, X) = 0, \forall X \in T_{p} M} .

(2.1)

The complementary vector bundle

S (T M)

Rad T M

in TM is called the screen bundle of M. We note that any screen bundle is non-degenerate. Thus, we have

T M = Rad T M ⊥ S (T M),

(2.2)

where ⊥ denotes the orthogonal direct sum. The complementary vector bundle

S {(T M)}^{⊥}

S (T M)

is called screen transversal bundle and it has rank 2. Since

Rad T M

is a lightlike subbundle of

S {(T M)}^{⊥}

, there exists a unique local section N of

S {(T M)}^{⊥}

such that

\tilde{g} (N, N) = 0, \tilde{g} (N, ξ) = 1 .

(2.3)

The Gauss and Weingarten formulas are given, respectively, by

\begin{aligned} {\tilde{\nabla}}_{X} Y = \nabla_{X} Y + h (X, Y), \\ {\tilde{\nabla}}_{X} N = - A_{N} X + \nabla_{X}^{t} N \end{aligned}

(2.4)

for any

X, Y \in Γ (T M)

, where

\nabla_{X} Y, A_{N} X \in Γ (T M)

and

h (X, Y), \nabla_{X}^{t} N \in Γ (ltr (T M))

. If we put

B (X, Y) = \tilde{g} (h (X, Y), ξ)

and

τ (X) = \tilde{g} (\nabla_{X}^{t} N, ξ)

, then (2.4) become

\begin{aligned} {\tilde{\nabla}}_{X} Y = \nabla_{X} Y + B (X, Y) N, \\ {\tilde{\nabla}}_{X} N = - A_{N} X + τ (X) N, \end{aligned}

(2.5)

where B and

A_{N}

are called the second fundamental form and the shape operator of the lightlike hypersurface M. The induced connection ∇ on M is not metric connection but ∇ is torsion free [21].

B = 0

, then the lightlike hypersurface M is called totally geodesic in

\tilde{M}

. A point

p \in M

is said to be umbilical if

B {(X, Y)}_{p} = H g_{p} (X, Y), \forall X, Y \in T_{p} M,

where

H \in R

. M is called totally umbilical in

\tilde{M}

if every point of M is umbilical [21].

The mean curvature μ of M with respect to an

{e_{1}, \dots, e_{n}}

orthonormal basis of

Γ (S (T M))

is defined in [22] as follows:

μ = \frac{tr (B)}{n} = \frac{1}{n} \sum_{i = 1}^{n} ε_{i} B (e_{i}, e_{i}), g (e_{i}, e_{i}) = ε_{i} .

(2.6)

Let P be a projection of

S (T M)

on M. From (2.2), we have

\begin{aligned} \nabla_{X} P Y & = \nabla_{X}^{*} P Y + h (X, P Y) \\ = \nabla_{X}^{*} Y + C (X, P Y) ξ, X, Y \in Γ (T M), \end{aligned}

(2.7)

\begin{aligned} \nabla_{X} ξ = - A_{ξ}^{*} X - τ (X) ξ, \end{aligned}

(2.8)

where

\nabla_{X}^{*} P Y

and

A_{ξ}^{*} X

belong to

Γ (S (T M))

. Here

\nabla^{*}

, C and

A_{ξ}^{*}

are called the induced connection, the local second fundamental form and the local shape operator on

S (T M)

, respectively.

From (2.5) and (2.7) one has

B (X, Y) = g (A_{ξ}^{*} X, Y),

(2.9)

C (X, P Y) = g (A_{N} X, P Y) .

(2.10)

Using (2.9) we have

B (X, ξ) = 0, X \in Γ (T M |_{U}) .

A lightlike hypersurface

(M, g)

of a semi-Riemannian manifold

(\tilde{M}, \tilde{g})

is called screen locally conformal if the shape operators

A_{N}

and

A_{ξ}^{*}

of M and

S (T M)

, respectively, are related by

A_{N} = φ A_{ξ}^{*},

(2.11)

where φ is a non-vanishing smooth function on a neighborhood U on M [23]. In particular, M is called screen homothetic if φ is a non-zero constant.

We denote the Riemann curvature tensors of

\tilde{M}

and M by

\tilde{R}

and R, respectively. The Gauss-Codazzi type equations for M are given as follows:

\begin{aligned} \tilde{g} (\tilde{R} (X, Y) Z, P U) = & g (R (X, Y) Z, P U) + B (X, Z) C (Y, P U) \\ - B (Y, Z) C (X, P U), \end{aligned}

(2.12)

\begin{aligned} \tilde{g} (\tilde{R} (X, Y) Z, ξ) = (\nabla_{X} B) (Y, Z) - (\nabla_{Y} B) (X, Z) + B (Y, Z) τ (X) - B (X, Z) τ (Y), \end{aligned}

(2.13)

\begin{aligned} \tilde{g} (\tilde{R} (X, Y) Z, N) = g (R (X, Y) Z, N), \end{aligned}

(2.14)

\begin{aligned} \tilde{g} (\tilde{R} (X, Y) P Z, N) = & (\nabla_{X} C) (Y, P Z) - (\nabla_{Y} C) (X, P Z) \\ + τ (Y) C (X, P Z) - τ (X) C (Y, P Z) \end{aligned}

(2.15)

for any

X, Y, Z, U \in Γ (T M)

[21].

Let

p \in M

and

Π = sp {e_{i}, e_{j}}

be a two-dimensional non-degenerate plane of

T_{p} M

. The number

K_{i j} = \frac{g (R (e_{j}, e_{i}) e_{i}, e_{j})}{g (e_{i}, e_{i}) g (e_{j}, e_{j}) - g {(e_{i}, e_{j})}^{2}}

is called the sectional curvature at

p \in M

. Since the screen second fundamental form C is not symmetric, the sectional curvature

K_{i j}

of a lightlike submanifold is not symmetric, that is,

K_{i j} \neq K_{j i}

Let

p \in M

and ξ be a null vector of

T_{p} M

. A plane Π of

T_{p} M

is called a null plane if it contains ξ and

e_{i}

such that

\tilde{g} (ξ, e_{i}) = 0

and

\tilde{g} (e_{i}, e_{i}) \neq 0

. The null sectional curvature of Π is given in [24] as follows:

K_{i}^{null} = \frac{g (R_{p} (e_{i}, ξ) ξ, e_{i})}{g_{p} (e_{i}, e_{i})} .

The Ricci tensor

\tilde{Ric}

\tilde{M}

and the induced Ricci type tensor

R^{(0, 2)}

of M are defined by

\begin{aligned} \tilde{Ric} (X, Y) = trace {Z \to \tilde{R} (Z, X) Y}, \forall X, Y \in Γ (T \tilde{M}), \\ R^{(0, 2)} (X, Y) = trace {Z \to R (Z, X) Y}, \forall X, Y \in Γ (T M) . \end{aligned}

(2.16)

Let

{e_{1}, \dots, e_{n}}

be an orthonormal frame of

Γ (S (T M))

. In this case,

R^{(0, 2)} (X, Y) = \sum_{j = 1}^{n} ε_{j} g (R (e_{j}, X) Y, e_{j}) + \tilde{g} (R (ξ, X) Y, N),

(2.17)

where

ε_{j}

denotes the causal character (∓1) of a vector field

e_{j}

. Ricci curvature of a lightlike hypersurface is not symmetric. Thus, Einstein hypersurfaces are not defined on any lightlike hypersurface. If M admits that an induced symmetric Ricci tensor Ric and Ricci tensor satisfy

Ric (X, Y) = k g (X, Y), \forall X, Y \in Γ (T M),

(2.18)

where k is a constant, then M is called an Einstein hypersurface [23].

Let M be a lightlike hypersurface of a Lorentzian manifold

\tilde{M}

, replacing X by ξ and using (2.12), (2.13) and (2.14)

\begin{aligned} R^{(0, 2)} (ξ, ξ) & = \sum_{j = 1}^{n} g (R (e_{j}, ξ) ξ, e_{j}) - \tilde{g} (R (ξ, ξ) ξ, N) \\ = \sum_{j = 1}^{n} K_{j}^{null} . \end{aligned}

(2.19)

Thus, we have

\sum_{i = 1}^{n} R^{(0, 2)} (e_{i}, e_{i}) = \sum_{i = 1}^{n} {\sum_{j = 1}^{n} g (R (e_{j}, e_{i}) e_{i}, e_{j})} + \sum_{i = 1}^{n} \tilde{g} (R (ξ, e_{i}) e_{i}, N) .

(2.20)

Adding (2.19) and (2.20), we obtain a scalar τ given as follows [25]:

\begin{aligned} τ & = R^{(0, 2)} (ξ, ξ) + \sum_{i = 1}^{n} R^{(0, 2)} (e_{i}, e_{i}) \\ = \sum_{i, j = 1}^{n} K_{i j} + \sum_{i = 1}^{n} K_{i}^{null} + K_{i N}, \end{aligned}

(2.21)

where

K_{i N} = \tilde{g} (R (ξ, e_{i}) e_{i}, N)

for

i \in {1, \dots, n}

3 k-Ricci curvature and k-scalar curvature

Let M be an

(n + 1)

-dimensional lightlike hypersurface of a Lorentzian manifold

\tilde{M}

and let

{e_{1}, \dots, e_{n}, ξ}

be a basis of

Γ (T M)

where

{e_{1}, \dots, e_{n}}

is an orthonormal basis of

Γ (S (T M))

. For

k \leq n

, we set

π_{k, ξ} = sp {e_{1}, \dots, e_{k}, ξ}

is a

(k + 1)

-dimensional degenerate plane section and

π_{k} = sp {e_{1}, \dots, e_{k}}

is a k-dimensional non-degenerate plane section.

We say that k-degenerate Ricci curvature and k-Ricci curvature at unit vector

X \in Γ (T M)

are as follows:

{Ric}_{π_{k, ξ}} (X) = R^{(0, 2)} (X, X) = \sum_{j = 1}^{k} g (R (e_{j}, X) X, e_{j}) + \tilde{g} (R (ξ, X) X, N),

(3.1)

{Ric}_{π_{k}} (X) = R^{(0, 2)} (X, X) = \sum_{j = 1}^{k} g (R (e_{j}, X) X, e_{j}),

(3.2)

respectively. Furthermore, we say that k-degenerate scalar curvature and k-scalar curvature at

p \in M

are as follows:

τ_{π_{k, ξ}} (p) = \sum_{i, j = 1}^{k} K_{i j} + \sum_{i = 1}^{k} K_{i}^{null} + K_{i N},

(3.3)

τ_{π_{k}} (p) = \sum_{i, j = 1}^{k} K_{i j},

(3.4)

respectively. For

k = 2

Π_{1, ξ} = sp {e_{1}, ξ}

, then we have

{Ric}_{Π_{1, ξ}} (e_{1}) = K_{1 N},

and

τ_{Π_{2}} (p) = K_{1}^{null} + K_{1 N} .

For

k = n

π_{n} = sp {e_{1}, \dots, e_{n}} = Γ (S (T M))

, then

{Ric}_{S (T M)} (e_{1}) = {Ric}_{π_{n}} (e_{1}) = \sum_{j = 1}^{n} K_{1 j} = K_{12} + \dots + K_{1 n},

(3.5)

and

τ_{S (T M)} (p) = \sum_{i, j = 1}^{n} K_{i j} .

(3.6)

We say that screen Ricci curvature and screen scalar curvature are

{Ric}_{S (T M)} (e_{1})

and

τ_{S (T M)} (p)

, respectively. From (2.12) we can write

τ_{S (T M)} (p) = {\tilde{τ}}_{S (T M)} (p) + \sum_{i, j = 1}^{n} B_{i i} C_{j j} - B_{i j} C_{j i},

(3.7)

where

B_{i j} = B (e_{i}, e_{j})

and

C_{i j} = C (e_{i}, e_{j})

for

i, j \in {1, \dots, n}

Also, the components of the second fundamental form B and the screen second fundamental form C satisfy

\sum_{i, j = 1}^{n} B_{i j} C_{j i} = \frac{1}{2} {\sum_{i, j = 1}^{n} {(B_{i j} + C_{j i})}^{2} - \sum_{i, j = 1}^{n} {(B_{i j})}^{2} + {(C_{j i})}^{2}},

(3.8)

and

\sum_{i, j} B_{i i} C_{j j} = \frac{1}{2} {{(\sum_{i, j} B_{i i} + C_{j j})}^{2} - {(\sum_{i} B_{i i})}^{2} - {(\sum_{j} C_{j j})}^{2}} .

(3.9)

Theorem 3.1 Let M be an

(n + 1)

-dimensional lightlike hypersurface of a Lorentzian manifold

\tilde{M}

. Then:

(a)

τ_{S (T M)} (p) \leq {\tilde{τ}}_{S (T M)} (p) + n μ (trace A_{N}) + \frac{1}{4} \sum_{i, j} {(B_{i j} - C_{j i})}^{2} .

(3.10)

The equality holds for all

p \in M

if and only if either M is a screen homothetic lightlike hypersurface with

φ = - 1

or M is a totally geodesic lightlike hypersurface.

(b)

τ_{S (T M)} (p) \geq {\tilde{τ}}_{S (T M)} (p) + n μ (trace A_{N}) - \frac{1}{2} \sum_{i, j} {(B_{i j} + C_{j i})}^{2} .

(3.11)

The equality holds for all

p \in M

if and only if either M is a screen homothetic lightlike hypersurface with

φ = 1

or M is a totally geodesic lightlike hypersurface.

(c)

The equalities case of (3.10) and (3.11) hold at

p \in M

if and only if p is a totally geodesic point.

Proof Using (3.7) and (3.8), we get

τ_{S (T M)} (p) = {\tilde{τ}}_{S (T M)} (p) + \sum_{i, j = 1}^{n} B_{i i} C_{j j} - \frac{1}{2} \sum_{i, j = 1}^{n} {(B_{i j} + C_{j i})}^{2} + \frac{1}{2} \sum_{i, j = 1}^{n} {(B_{i j})}^{2} + {(C_{j i})}^{2} .

(3.12)

Since

\frac{1}{2} (B_{i j}^{2} + C_{j i}^{2}) = \frac{1}{4} {(B_{i j} + C_{j i})}^{2} + \frac{1}{4} {(B_{i j} - C_{j i})}^{2},

then

\begin{aligned} \frac{1}{2} {\sum_{i, j = 1}^{n} {(B_{i j} + C_{j i})}^{2} + \sum_{i, j = 1}^{n} {(B_{i j})}^{2} + {(C_{j i})}^{2}} \\ = - \frac{1}{2} \sum_{i, j} {(B_{i j} + C_{j i})}^{2} + \frac{1}{4} \sum_{i, j} {(B_{i j} - C_{j i})}^{2} . \end{aligned}

(3.13)

If we put (3.13) in (3.12), we obtain

τ_{S (T M)} (p) = {\tilde{τ}}_{S (T M)} (p) + \sum_{i, j = 1}^{n} B_{i i} C_{j j} - \frac{1}{2} \sum_{i, j} {(B_{i j} + C_{j i})}^{2} + \frac{1}{4} \sum_{i, j} {(B_{i j} - C_{j i})}^{2},

(3.14)

which yields (3.10) and (3.11). From (3.10), (3.11) and (3.14) it is easy to get (a), (b) and (c) statements. □

Corollary 3.2 Let M be an

(n + 1)

-dimensional lightlike hypersurface of a Lorentzian space form

\tilde{M} (c)

. Then:

(a)

τ_{S (T M)} (p) \leq n (n - 1) c + n μ (trace A_{N}) + \frac{1}{4} \sum_{i, j} {(B_{i j} - C_{j i})}^{2} .

(3.15)

(b)

τ_{S (T M)} (p) \geq n (n - 1) c + n μ (trace A_{N}) - \frac{1}{2} \sum_{i, j} {(B_{i j} + C_{j i})}^{2} .

(3.16)

Corollary 3.3 Let M be an

(n + 1)

-dimensional screen homothetic lightlike hypersurface of a Lorentzian space form

\tilde{M} (c)

. Then:

(a)

τ_{S (T M)} (p) \leq n (n - 1) c + φ n^{2} μ^{2} + \frac{(φ - 1)}{4} \sum_{i, j} {(B_{i j})}^{2} .

(3.17)

(b)

τ_{S (T M)} (p) \geq n (n - 1) c + φ n^{2} μ^{2} - \frac{(φ - 1)}{2} \sum_{i, j} {(B_{i j})}^{2} .

(3.18)

Theorem 3.4 Let M be an

(n + 1)

-dimensional lightlike hypersurface of a Lorentzian manifold

\tilde{M}

. Then

\begin{aligned} τ_{S (T M)} (p) \leq & {\tilde{τ}}_{S (T M)} (p) + \frac{1}{2} {(trace \bar{A})}^{2} - \frac{1}{2} {(trace A_{N})}^{2} \\ - \frac{1}{2} \sum_{i, j} {(B_{i j} + C_{j i})}^{2} + \frac{1}{4} \sum_{i, j} {(B_{i j} - C_{j i})}^{2}, \end{aligned}

(3.19)

where

\bar{A} = (\begin{array}{c} B_{11} + C_{11} & B_{12} + C_{21} & \dots & B_{1 n} + C_{n 1} \\ B_{21} + C_{12} & B_{22} + C_{22} & \dots & B_{2 n} + C_{n 2} \\ ⋮ \\ B_{n 1} + C_{1 n} & B_{n 2} + C_{2 n} & \dots & B_{n n} + C_{n n} \end{array}) .

(3.20)

The equality of (3.19) holds for all

p \in M

if and only if M is minimal.

Proof From (3.14) and (3.9) we get

\begin{aligned} τ_{S (T M)} (p) = & {\tilde{τ}}_{S (T M)} (p) + \frac{1}{2} {{(\sum_{i, j} B_{i i} + C_{j j})}^{2} - {(\sum_{i} B_{i i})}^{2} - {(\sum_{j} C_{j j})}^{2}} \\ - \frac{1}{2} \sum_{i, j} {(B_{i j} + C_{j i})}^{2} + \frac{1}{4} \sum_{i, j} {(B_{i j} - C_{j i})}^{2}, \end{aligned}

(3.21)

which implies (3.19).

The equality of (3.19) satisfies then

\sum_{i} B_{i i} = 0 .

(3.22)

This shows that M is minimal. □

By Theorem 3.4 we get the following corollaries.

Corollary 3.5 Let M be an

(n + 1)

-dimensional lightlike hypersurface of a Lorentzian space form

\tilde{M} (c)

. Then

\begin{aligned} τ_{S (T M)} (p) \leq & n (n - 1) c + \frac{1}{2} {(trace \bar{A})}^{2} - \frac{1}{2} {(trace A_{N})}^{2} \\ - \frac{1}{2} \sum_{i, j} {(B_{i j} + C_{j i})}^{2} + \frac{1}{4} \sum_{i, j} {(B_{i j} - C_{j i})}^{2}, \end{aligned}

(3.23)

where

\bar{A}

is equal to (3.20). The equality of (3.23) holds for all

p \in M

if and only if M is minimal.

Corollary 3.6 Let M be an

(n + 1)

-dimensional screen homothetic lightlike hypersurface of a Lorentzian manifold

\tilde{M}

. Then

τ_{S (T M)} (p) \leq {\tilde{τ}}_{S (T M)} (p) + \frac{(2 φ + 1)}{2} n^{2} μ^{2} - \frac{(φ^{2} + 6 φ + 1)}{4} \sum_{i, j} {(B_{i j})}^{2} .

(3.24)

The equality of (3.24) holds for all

p \in M

if and only if M is minimal.

Now, we shall need the following lemma.

Lemma 3.7 [26]

a_{1}, \dots, a_{n}

are n-real numbers (

n > 1

), then

\frac{1}{n} {(\sum_{i = 1}^{n} a_{i})}^{2} \leq \sum_{i = 1}^{n} a_{i}^{2},

(3.25)

with equality if and only if

a_{1} = \dots = a_{n}

Theorem 3.8 Let M be an

(n + 1)

-dimensional (

n > 1

) lightlike hypersurface of a Lorentzian manifold

\tilde{M}

. Then

\begin{aligned} τ_{S (T M)} (p) \leq & {\tilde{τ}}_{S (T M)} (p) + \frac{n - 1}{2 n} {(trace \bar{A})}^{2} - \frac{1}{2} {{(trace A_{N})}^{2} + n^{2} μ^{2}} \\ + \frac{1}{4} \sum_{i, j} {(B_{i j} - C_{j i})}^{2} - \frac{1}{2} \sum_{i \neq j} {(B_{i j} + C_{j i})}^{2}, \end{aligned}

(3.26)

where

\bar{A}

is equal to (3.20).

The equality case of (3.26) holds for all

p \in M

if and only if

n μ = - trace A_{N}

Proof From (3.21)

\begin{aligned} τ_{S (T M)} (p) = & {\tilde{τ}}_{S (T M)} (p) + \frac{1}{2} {{(trace \bar{A})}^{2} - {(trace A_{N})}^{2} - n^{2} μ^{2}} - \frac{1}{2} \sum_{i} {(B_{i i} + C_{i i})}^{2} \\ - \frac{1}{2} \sum_{i \neq j} {(B_{i j} + C_{j i})}^{2} + \frac{1}{4} \sum_{i, j} {(B_{i j} - C_{j i})}^{2} . \end{aligned}

(3.27)

Using Lemma 3.7 and equality (3.27), we have

\begin{array}{rcl} τ_{S (T M)} (p) & \leq & {\tilde{τ}}_{S (T M)} (p) + \frac{1}{2} {{(trace \bar{A})}^{2} - {(trace A_{N})}^{2} - n^{2} μ^{2}} - \frac{1}{2 n} {(\sum_{i} B_{i i} + C_{i i})}^{2} \\ - \frac{1}{2} \sum_{i \neq j} {(B_{i j} + C_{j i})}^{2} + \frac{1}{4} \sum_{i, j} {(B_{i j} - C_{j i})}^{2}, \end{array}

(3.28)

which implies (3.26).

The equality case of (3.26) satisfies then

B_{11} + C_{11} = \dots = B_{n n} + C_{n n} .

(3.29)

From (3.29) we get

\begin{matrix} (1 - n) B_{11} + B_{22} + \dots + B_{n n} + (1 - n) C_{11} + C_{22} + \dots + C_{n n} = 0, \\ B_{11} + (1 - n) B_{22} + \dots + B_{n n} + C_{11} + (1 - n) C_{22} + \dots + C_{n n} = 0, \\ ⋮ \\ B_{11} + B_{22} + \dots + (1 - n) B_{n n} + C_{11} + C_{22} + \dots + (1 - n) C_{n n} = 0 . \end{matrix}

By the above equations, we obtain

{(n - 1)}^{2} \cdot (trace A_{N} + n μ) = 0 .

(3.30)

Since

n \neq 1

n μ = - trace A_{N}

. □

From Theorem 3.8 we get the following corollaries.

Corollary 3.9 Let M be an

(n + 1)

-dimensional (

n > 1

) lightlike hypersurface of a Lorentzian space form

\tilde{M} (c)

. Then

\begin{aligned} τ_{S (T M)} (p) \leq & n (n - 1) c + \frac{n - 1}{2 n} {(trace \bar{A})}^{2} - \frac{1}{2} {n^{2} μ^{2} + {(trace A_{N})}^{2}} \\ + \frac{1}{4} \sum_{i, j} {(B_{i j} - C_{j i})}^{2} - \frac{1}{2} \sum_{i \neq j} {(B_{i j} + C_{j i})}^{2}, \end{aligned}

(3.31)

where

\bar{A}

is equal to (3.20).

The equality case of (3.31) holds for all

p \in M

if and only if

n μ = - trace A_{N}

Corollary 3.10 Let M be an

(n + 1)

-dimensional (

n > 1

) screen homothetic lightlike hypersurface of a Lorentzian manifold

\tilde{M}

. Then

\begin{aligned} τ_{S (T M)} (p) \leq & {\tilde{τ}}_{S (T M)} (p) + φ n^{2} μ^{2} - \frac{{(φ + 1)}^{2}}{2} n μ^{2} \\ + \frac{{(φ - 1)}^{2}}{4} \sum_{i} {(B_{i i})}^{2} - \frac{(φ^{2} + 6 φ + 1)}{4} \sum_{i \neq j} {(B_{i j})}^{2} . \end{aligned}

(3.32)

The equality case of (3.32) holds for all

p \in M

if and only if either

φ = - 1

or M is minimal.

4 Curvature invariants on lightlike hypersurfaces

Definition 4.1 For an integer

k \geq 0

, let

S (n, k)

be the finite set which consists of k-tuples

(n_{1}, \dots, n_{k})

of integers ≥2 satisfying

n_{1} < n

and

n_{1} + \dots + n_{k} \leq n

. Denote by

S (n)

the set of all unordered k-tuples with

k \geq 0

for a fixed positive integer n.

For each k-tuple

(n_{1}, \dots, n_{k}) \in S (n)

, the two sequences of curvature invariants

S (n_{1}, \dots, n_{k}) (p)

and

\hat{S} (n_{1}, \dots, n_{k}) (p)

are defined by, respectively,

\begin{matrix} S (n_{1}, \dots, n_{k}) (p) = inf {τ (Π_{n_{1}}) + \dots + τ (Π_{n_{k}})}, \\ \hat{S} (n_{1}, \dots, n_{k}) (p) = sup {τ (Π_{n_{1}}) + \dots + τ (Π_{n_{k}})}, \end{matrix}

where

Π_{n_{1}}, \dots, Π_{n_{k}}

are k-dimensional mutually orthogonal subspaces of

T_{p} M

such that

dim Π_{n_{j}} = n_{j}

j = 1, \dots, k

\begin{matrix} δ (n_{1}, \dots, n_{k}) (p) = τ (p) - S (n_{1}, \dots, n_{k}) (p), \\ \hat{δ} (n_{1}, \dots, n_{k}) (p) = τ (p) - \hat{S} (n_{1}, \dots, n_{k}) (p) . \end{matrix}

We call a lightlike hypersurface an

S (n_{1}, \dots, n_{k})

space if it satisfies

S (n_{1}, \dots, n_{k}) = \hat{S} (n_{1}, \dots, n_{k})

Theorem 4.2 Let M be a lightlike hypersurface of an

(n + 2)

-dimensional Lorentzian manifold

\tilde{M}

. Then M is an

S (n)

space if and only if the scalar curvature of M is constant.

Proof Let

{e_{1}, \dots, e_{n}}

be an orthonormal frame of

Γ (S (T M))

. Let us choose n-dimensional plane sections such that

\begin{matrix} π_{n - 1, ξ}^{1} = sp {e_{2}, \dots, e_{n}, ξ} \subset T_{p} M, \\ ⋮ \\ π_{n - 1, ξ}^{n} = sp {e_{1}, \dots, e_{n - 1}, ξ} \subset T_{p} M, \\ π_{n} = sp {e_{1}, \dots, e_{n}} \subset T_{p} M . \end{matrix}

Thus, from (3.3) and (3.4), we obtain

\begin{matrix} τ_{π_{n - 1, ξ}^{1}} (p) = \sum_{i, j = 2}^{n} K_{i j} + K_{i}^{null} + K_{i N}, \\ ⋮ \\ τ_{π_{n - 1, ξ}^{n}} (p) = \sum_{i, j = 1}^{n - 1} K_{i j} + K_{i}^{null} + K_{i N}, \\ τ_{π_{n}} (p) = \sum_{i, j = 1}^{n} K_{i j} = τ_{S (T M)} (p) . \end{matrix}

If M is an

S (n)

space, then we can write

\begin{matrix} τ_{π_{n - 1, ξ}^{1}} (p) = τ (p) - Ric (e_{1}) - K_{1}^{null} = c, \\ ⋮ \\ τ_{π_{n - 1, ξ}^{n}} (p) = τ (p) - Ric (e_{n}) - K_{n}^{null} = c, \\ τ_{π_{n}} (p) = τ (p) - \sum_{i = 1}^{n} K_{i}^{null} + K_{i N} = c . \end{matrix}

Therefore, we have

Ric (e_{1}) + K_{1}^{null} = \dots = Ric (e_{n}) + K_{n}^{null} = \sum_{i = 1}^{n} K_{i}^{null} + K_{i N} .

(4.1)

From (4.1) we get

{Ric}_{S (T M)} (e_{1}) + \dots + {Ric}_{S (T M)} (e_{n}) = (n - 1) \sum_{i = 1}^{n} K_{i}^{null} + K_{i N} .

(4.2)

Using (4.2) we have

τ_{S (T M)} (p) = (n - 1) (τ (p) - τ_{S (T M)} (p)) .

Thus, we obtain

τ (p) = (\frac{2 - n}{1 - n}) c,

(4.3)

which shows that

τ (p)

is constant, which completes the proof. □

Remark 4.3 We note that if an n-dimensional non-degenerate manifold is an

S (n)

space, then it is an Einstein space (see [10]). On the other hand, if a degenerate hypersurface of a lightlike hypersurface is an

S (n)

space, then it has constant scalar curvature. Thus, the curvature invariants on degenerate submanifolds give different characterizations from the curvature invariants on non-degenerate submanifolds.

Keeping in view (4.2), we get the following corollary immediately.

Corollary 4.4 Let

M (c)

be an n-dimensional lightlike hypersurface with constant sectional curvature c.

M (c)

is an

S (n)

space if and only if

\sum_{i = 1}^{n} K_{i N} = 0

Now, we prove the following.

Theorem 4.5 Let M be a lightlike hypersurface of an

(n + 2)

-dimensional Lorentzian manifold

\tilde{M}

. If M is an

S (j)

space for

2 \leq j < n

, then M is also

S (j + 1)

space.

Proof For the proof of the theorem, we use the induction method. Firstly, we show that the claim of the theorem is true for

j = 2

. Suppose that M is an

S (2)

space. Let us choose any two-dimensional plane sections of

T_{p} M

Π_{1, ξ}^{1} = sp {e_{1}, ξ}

Π_{2} = {e_{1}, e_{2}}

Π_{1, ξ}^{2} = sp {e_{2}, ξ}

. In that case,

\begin{matrix} τ_{Π_{1, ξ}^{1}} (p) = K_{1}^{null} + K_{1 N} = c, \\ τ_{Π_{2}} (p) = K_{12} + K_{21} = c, \\ τ_{Π_{1, ξ}^{2}} (p) = K_{2}^{null} + K_{2 N} = c . \end{matrix}

Now, let us choose three-dimensional plane sections of

T_{p} M

π_{3} = sp {e_{1}, e_{2}, e_{3}}

π_{2, ξ}^{1} = sp {e_{1}, e_{2}, ξ}

. If we show that

τ_{π_{2, ξ}^{1}} (p) = τ_{π_{3}} (p) = constant

, then M is an

S (3)

-space

\begin{aligned} τ_{π_{2, ξ}^{1}} (p) & = K_{12} + K_{21} + \sum_{i = 1}^{2} K_{i}^{null} + K_{i N} \\ = 3 c, \end{aligned}

and

\begin{array}{rcl} τ_{π_{3}} (p) & = & K_{12} + K_{21} + K_{13} + K_{31} + K_{23} + K_{32} \\ = & 3 c . \end{array}

Therefore, M is an

S (3)

space.

Now, we show that the claim of the theorem is true for

n = k

Let us choose any k-dimensional plane sections of

T_{p} M

π_{k - 1, ξ}^{1} = sp {e_{2}, e_{3}, \dots, e_{k}, ξ}

π_{k - 1, ξ}^{2} = sp {e_{1}, e_{3}, \dots, e_{k}, ξ}

, …,

π_{k - 1, ξ}^{k} = sp {e_{1}, e_{2}, \dots, e_{k - 1}, ξ}

π_{k} = sp {e_{1}, e_{2}, \dots, e_{k}}

. Then

\begin{matrix} τ_{π_{k - 1, ξ}^{1}} (p) = \sum_{i, j = 2}^{k} K_{i j} + \sum_{i = 2}^{k} K_{i}^{null} + K_{i N}, \\ ⋮ \\ τ_{π_{k - 1, ξ}^{k}} (p) = \sum_{i, j = 1}^{k - 1} K_{i j} + \sum_{i = 1}^{k - 1} K_{i}^{null} + K_{i N}, \\ τ_{π_{k}} (p) = \sum_{i, j = 1}^{k} K_{i j} . \end{matrix}

From the above equations, we have

\sum_{i = 1}^{k} K_{i}^{null} + K_{i N} = \frac{2 c}{k - 1} .

Let us choose

(k + 1)

-dimensional plane sections of

T_{p} M

π_{k, ξ} = sp {e_{1}, \dots, e_{k}, ξ}

π_{k + 1} = sp {e_{1}, \dots, e_{k}, e_{k + 1}}

, then

\begin{aligned} τ_{π_{k, ξ}} (p) & = \sum_{i, j = 1} K_{i j} + \sum_{i = 1}^{k} K_{i}^{null} + K_{i N} \\ = c + \frac{2 c}{k - 1} = (\frac{k + 1}{k - 1}) c . \end{aligned}

(4.4)

Using in a similar way a special case

j = 2

, we obtain

τ_{π_{k + 1}} (p) = (\frac{k + 1}{k - 1}) c .

(4.5)

From (4.4) and (4.5) M is an

S (k + 1)

space. □

Theorem 4.6 Let M be a lightlike hypersurface of an

(n + 2)

-dimensional Lorentzian manifold

\tilde{M}

. Let

{e_{1}, \dots, e_{n}, ξ}

be an orthonormal basis of

p \in M

. If M is an

S (n - 1)

space, then

Ric (e_{1}) = \dots = Ric (e_{n}) = constant

and

K_{1}^{null} = \dots = K_{n}^{null}

Proof Let M be an

S (n - 1)

space and

π_{n - 2, ξ}^{1} = sp {e_{2}, \dots, e_{n - 1}, ξ}

π_{n - 2, ξ}^{2} = sp {e_{1}, e_{3}, \dots, e_{n - 1}, ξ}

,…,

π_{n - 2, ξ}^{n - 1} = sp {e_{1}, e_{2}, \dots, e_{n - 2}, ξ}

π_{n - 1} = sp {e_{1}, e_{2}, \dots, e_{n - 1}}

(n - 1)

-dimensional plane sections of

T_{p} M

. Then

\begin{matrix} τ_{π_{n - 2, ξ}^{1}} (p) = τ (p) - Ric (e_{1}) - Ric (e_{n}) - K_{1}^{null} - K_{n}^{null} = c, \\ τ_{π_{n - 2, ξ}^{2}} (p) = τ (p) - Ric (e_{2}) - Ric (e_{n}) - K_{1}^{null} - K_{n}^{null} = c, \\ ⋮ \\ τ_{π_{n - 2, ξ}^{n - 1}} (p) = τ (p) - Ric (e_{n - 1}) - Ric (e_{n}) - K_{n - 1}^{null} - K_{n}^{null} = c, \\ τ_{π_{n - 1}} (p) = τ (p) - Ric (e_{n}) - \sum_{i = 1}^{n} K_{i}^{null} + K_{i N} = c . \end{matrix}

If we sum the above equations side to side and take into consideration Theorem 4.5, we have

Ric (e_{1}) + \dots + Ric (e_{n}) + (n - 1) Ric (e_{n}) + \sum_{i = 1}^{n} K_{i}^{null} + \sum_{i = 1}^{n - 1} K_{i}^{null} + K_{i N} = constant .

Therefore, we obtain

\begin{matrix} {Ric}_{S (T M)} (e_{1}) + \dots + {Ric}_{S (T M)} (e_{n}) + (n - 1) Ric (e_{n}) + \sum_{i = 1}^{n} K_{i}^{null} + K_{i N} + \sum_{i = 1}^{n - 1} K_{i}^{null} + K_{i N} \\ = constant . \end{matrix}

Taking into account upper equations, we get

τ_{S (T M)} (p) + (n - 1) Ric (e_{n}) + τ (p) - τ_{S (T M)} (p) + τ_{π_{n - 1, ξ}} (p) - τ_{π_{n - 1}} (p) = constant,

where

π_{n - 1, ξ} = sp {e_{1}, \dots, e_{n - 1}, ξ}

and

π_{n - 1} = sp {e_{1}, \dots, e_{n - 1}}

. Using Theorem 4.2 and Theorem 4.5, we obtain

Ric (e_{n}) = constant

. In addition to this, from (4.1), Theorem 4.2 and Theorem 4.5, we have

K_{1}^{null} = \dots = K_{n}^{null}

, which completes the proof of the theorem. □

In [25], Duggal restricted a lightlike hypersurface M (labeled by

M^{0}

) of genus zero with screen distribution

S {(T M)}^{0}

. He denoted this type of a lightlike hypersurface by

C [M^{0}] = [(M^{0}, g^{0}, S {(T M)}^{0})]

a class of lightlike hypersurfaces of genus zero such that

(a)

M^{0}

admits a canonical screen distribution

S {(T M)}^{0}

that induces a canonical lightlike transversal vector bundle

N^{0}

(b)

M^{0}

admits an induced symmetric Ricci tensor, denoted by Ric⁰.

From above information, we get the following theorem immediately.

Theorem 4.7 Let

M^{0}

, a member of

C [M^{0}]

, be an

(2 n + 1)

-dimensional lightlike hypersurface of a Lorentzian manifold

\tilde{M}

. If

M^{0}

is an Einstein hypersurface, then

τ_{π_{n, ξ}} (p) - τ_{π_{n, ξ}^{⊥}} (p) = \sum_{i = 1}^{n} K_{i}^{null} - \sum_{i = n + 1}^{2 n} K_{i}^{null},

(4.6)

where

π_{n, ξ}

is any

(n + 1)

-dimensional null section of

T_{p} M^{0}

and

π_{n, ξ}^{⊥}

denotes the orthogonal complement

π_{n, ξ}

T_{p} M^{0}

Proof Let us choose an orthonormal basis

{e_{1}, \dots, e_{2 n}}

at p such that

π_{n, ξ}

is spanned by

{e_{1}, \dots, e_{n}, ξ}

. If

M^{0}

is an Einstein hypersurface, then the Ricci curvature of

M^{0}

satisfies

Ric (e_{1}) + \dots + Ric (e_{n}) = Ric (e_{n + 1}) + \dots + Ric (e_{2 n}) .

From (2.17) we have

\sum_{i = 1}^{n} \sum_{j = 1}^{2 n} K_{i j} + \sum_{i = 1}^{n} K_{i N} = \sum_{i = n + 1}^{2 n} \sum_{j = 1}^{2 n} K_{i j} + \sum_{i = n + 1}^{2 n} K_{i N},

so, we get

\sum_{i, j = 1}^{n} K_{i j} + \sum_{i = 1}^{n} K_{i N} = \sum_{i, j = n + 1}^{n} K_{i j} + \sum_{i = n + 1}^{n} K_{i N},

which is equivalent to (4.6). □

Now, we introduce these invariants as some special cases, and we get interesting characterizations on lightlike hypersurfaces as follows.

Theorem 4.8 Let M be an

S (n_{1}, n_{1})

-space. Then:

(a)

n_{1} = 2

, then M is an

S (3)

-space.

(b)

n_{1} \neq 2

, then M is not necessary an

S (n_{1} + 1)

-space. If

\sum_{i = 1}^{n_{1}} K_{i}^{null} + K_{i N} = constant,

then M is an

S (n_{1} + 1)

-space.

Proof (a)

n_{1} = 2

, let us choose any two-dimensional plane sections of

T_{p} M

Π_{1, ξ}^{1} = sp {e_{1}, ξ}

Π_{1, ξ}^{2} = sp {e_{2}, ξ}

Π_{2} = sp {e_{1}, e_{2}}

. Then

\begin{matrix} τ_{Π_{1, ξ}^{1}} (p) = K_{1}^{null} + K_{1 N}, \\ τ_{Π_{1, ξ}^{2}} (p) = K_{2}^{null} + K_{2 N}, \\ τ_{Π_{2}} (p) = K_{12} + K_{21} . \end{matrix}

If M is an

S (2, 2)

space, then

\begin{matrix} τ_{Π_{1, ξ}^{1}} (p) + τ_{Π_{1, ξ}^{2}} (p) = \sum_{i = 1}^{2} K_{i}^{null} + K_{i N} = c, \\ τ_{Π_{1, ξ}^{1}} (p) + τ_{Π_{2}} (p) = K_{12} + K_{21} + K_{1}^{null} + K_{1 N} = c, \\ τ_{Π_{1, ξ}^{2}} (p) + τ_{Π_{2}} (p) = K_{12} + K_{21} + K_{2}^{null} + K_{2 N} = c . \end{matrix}

From the above equations, we have

τ_{π_{2, ξ}} (p) = \frac{3 c}{2},

(4.7)

where

π_{2, ξ} = sp {e_{1}, e_{2}, ξ}

is a three-dimensional null plane section of

T_{p} M

Now, let us choose any two-dimensional plane sections of

T_{p} M

Π_{2}^{1} = sp {e_{1}, e_{2}}

Π_{2}^{2} = sp {e_{1}, e_{3}}

Π_{2}^{3} = sp {e_{2}, e_{3}}

. Since M is an

S (2, 2)

-space, we can write

K_{12} + K_{21} + K_{13} + K_{31} + K_{23} + K_{32} + K_{12} + K_{21} + K_{13} + K_{31} + K_{23} + K_{32} = 2 τ (π_{3}) = 3 c .

Therefore,

τ (π_{4}) = \frac{3 c}{2},

(4.8)

where

π_{3} = sp {e_{1}, e_{2}, e_{3}}

is a three-dimensional non-degenerate plane section of

T_{p} M

. From (4.7) and (4.8) we see that M is an

S (3)

-space.

(b)

We show that the claim of the theorem is true for

n_{1} = 3

. Let us choose any three-dimensional plane section of

T_{p} M

π_{2, ξ}^{1} = sp {e_{1}, e_{2}, ξ}

π_{2, ξ}^{2} = sp {e_{2}, e_{3}, ξ}

π_{2, ξ}^{3} = sp {e_{1}, e_{3}, ξ}

. If M is an

S (3, 3)

-space, then

\begin{aligned} 3 c & = 2 (τ_{π_{2, ξ}^{1}} (p) + τ_{π_{2, ξ}^{2}} (p) + τ_{π_{2, ξ}^{3}} (p)) \\ = 2 τ_{π_{3, ξ}} (p) + 2 \sum_{i = 1}^{3} K_{i}^{null} + K_{i N}, \end{aligned}

(4.9)

where

π_{3, ξ} = sp {e_{1}, e_{2}, e_{3}, ξ} \subset T_{p} M

. Consider (4.9), we obtain the proof of (b) condition is true. □

The proof of a general case has been seen using the same way as the special case

n_{1} = 3

Theorem 4.9 Let M be a

(2 n + 1)

-dimensional lightlike hypersurface of a Lorentzian manifold

\tilde{M}

(a)

inf {τ (π_{n}) + τ (π_{n + 1})} > 0

, then

τ (p) > 0

(b)

sup {τ (π_{n}) + τ (π_{n + 1})} < 0

, then

τ (p) < 0

Proof Let

T_{p} M = sp {e_{1}, \dots, e_{2 n}, ξ}

. We suppose that

inf {τ (π_{n}) + τ (π_{n + 1})} > 0

. By straightforward computation, we have

C (2 n - 2, n - 2) + C (2 n - 2, n) τ_{S (T M)} (p) + C (2 n - 1, n - 1) \sum_{i = 1}^{2 n} K_{i}^{null} + K_{i N} > 0,

(4.10)

and

C (2 n - 2, n - 3) + C (2 n - 2, n - 1) τ_{S (T M)} (p) + C (2 n, n) \sum_{i = 1}^{2 n} K_{i}^{null} + K_{i N} > 0 .

(4.11)

Summing up (4.10) and (4.11), we get

C (2 n, n) τ_{S (T M)} (p) + C (2 n, n) \sum_{i = 1}^{2 n} K_{i}^{null} + K_{i N} > 0,

(4.12)

which shows that

C (2 n, n) τ (p) > 0

. Therefore,

τ (p) > 0

which is a proof of the statement (a).

Now, we suppose that

sup {τ (π_{n}) + τ (π_{n + 1})} > 0

. Following a similar way in the proof of statement (a), we have

C (2 n, n) τ_{S (T M)} (p) + C (2 n, n) \sum_{i = 1}^{2 n} K_{i}^{null} + K_{i N} < 0,

(4.13)

which shows that

C (2 n, n) τ (p) < 0

. Therefore

τ (p) < 0

, which is a proof of the statement (b). □

Acknowledgements

The authors have greatly benefited from the referee’s report. So we wish to express our gratitude to the reviewer for his/her valuable suggestions which improved the content and presentation of the paper.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors did not provide this information.

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Titel: Chen-like inequalities on lightlike hypersurfaces of a Lorentzian manifold
verfasst von: Mehmet Gülbahar
Erol Kilic̣
Sadık Keleṣ
Publikationsdatum: 01.12.2013
Verlag: Springer International Publishing
Erschienen in: Journal of Inequalities and Applications / Ausgabe 1/2013
Elektronische ISSN: 1029-242X
DOI: https://doi.org/10.1186/1029-242X-2013-266

Springer Professional

Chen-like inequalities on lightlike hypersurfaces of a Lorentzian manifold

Abstract

Competing interests

Authors’ contributions

1 Introduction

2 Preliminaries

3 k-Ricci curvature and k-scalar curvature

4 Curvature invariants on lightlike hypersurfaces

Acknowledgements

Competing interests

Authors’ contributions

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Springer Professional

Abstract

Competing interests

Authors’ contributions

1 Introduction

2 Preliminaries

3 k-Ricci curvature and k-scalar curvature

4 Curvature invariants on lightlike hypersurfaces

Acknowledgements

Competing interests

Authors’ contributions

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