#### Abstract

We study the problem of lightlike hypersurface immersed into Robertson-Walker (RW) spacetimes in this paper, where the screen bundle of the hypersurface has constant higher order mean curvature. We consider the following question: under what conditions is the compact lightlike hypersurface totally umbilical? Our approach is based on the relationship between the lightlike hypersurface with its screen bundle and the Minkowski formulae for the screen bundle.

#### 1. Introduction

The mathematical interests for the study of spacelike hypersurfaces in spacetimes began in the seventies with the works of Cheng and Yau [1], Brill and Flaherty [2], Choquet-Bruhat [3], and later on with some other authors, such as in [4–6]. Moreover, the study of such hypersurfaces is also of interest from a physical point of view, because of its relation to several problems in general relativity. More recently, there has been an increasing interest in the study of spacelike hypersurfaces with constant higher order mean curvature, such as in [7–9]. At the same time, in [10, 11] there are system research works about the lightlike hypersurface of semi-Riemannian manifolds. In this paper, based on the previous result, we want to make an attempt to study the lightlike hypersurfaces of spacetimes. First of all, we are interested in the study of lightlike hypersurfaces in conformally stationary spacetimes, and the screen bundle with constant higher order mean curvature.

First, we use the Newton transformations as the main analytical tool and study the Minkowski-type integral formulas of the lightlike hypersurface. The use of these kinds of formulas in the Lorentzian manifold was first started by Montiel in [12] for the spacelike hypersurface with constant mean curvature in de Sitter spacetimes, and it was continued by Alías et al. [13, 14] for more general spacetimes. Higher-order Minkowski formula for hypersurface was first obtained by Hsiung in [15] in Euclidean space, and by Bivens in [16] in the Euclidean sphere and hyperbolic space. In this paper, we obtain two Minkowski-type integral formulas about the higher-order mean curvature of the lightlike hypersurface as follows, which are called Minkowski formulas I and II.

*The Minkowski Formula I.* Let be a compact lightlike hypersurface of a conformally stationary spacetime of constant sectional curvature, on which the screen bundle is integrable and Ricci tensor of the induced connection is symmetric; then the following conclusion holds:

*The Minkowski Formula II.* Let be a compact lightlike hypersurface immersed into a conformally stationary spacetime of constant sectional curvature, where the screen bundle is integrable and the Ricci tensor of induced connection is symmetric; then

Second, by using the Minkowski-type integral formulas and divergence theorem, we get some sufficient conditions for lightlike hypersurfaces of conformally stationary spacetimes which are totally umbilical. The screen bundle of the lightlike hypersurface is spacelike; we study the lightlike hypersurfaces of conformally stationary spacetimes via the screen bundle and get our main results (cf. Theorems 10–13).

This paper is organized as follows. In Section 2, we state some preliminary knowledge related to this paper. In Section 3, we introduce the shape operator and the corresponding Newton transformations and derive some formulas about the divergence of Newton transformations. In Section 4, based on the results in Section 3, we obtain the Minkowski-type integral formulas I and II of lightlike hypersurfaces. In Section 5, as the application of Minkowski formulas, we obtain some sufficient conditions for lightlike hypersurfaces totally umbilical when the screen bundle is with constant higher order mean curvature.

#### 2. Preliminaries

Let be an -dimensional Riemannian manifold and let be a -dimensional manifold (either a circle or an open interval of ). We denote by the -dimensional product manifold endowed with the Lorentzian metric where is a positive smooth function on , and is the Riemannian metric on ; then we have that is a Lorentzian warped product with Lorentzian base , Riemannian fiber , , and warping function , when the sectional curvature of Riemannian factor is constant; then we call as Robertson-Walker (RW) spacetime.

Consider smooth immersion of an -dimensional connected lightlike manifold into a RW spacetime, where the induced metric on is ; then we have that is a lightlike hypersurface.

Let , be the unitary timelike conformal vector field globally defined on the RW spacetime ; then consider lightlike hypersurface ; there exists a pair lightlike vector fields defined on which satisfied the following formulas: There exists a unitary timelike vector field defined on which is the same time-orientation as ; we will refer to as the future-pointing Gauss map of the hypersurface; its opposite is the past-pointing Gauss map, where we can suppose on in the future-pointing.

Now we denote to be the connection of and , respectively, the projection morphism of onto ; then the Gauss and Weingarten formula for the hypersurface in are given as follows: where , , , and are the second fundamental forms of and , and are the local second fundamental forms on and , respectively, and are the local shape operator on and , respectively, and is the 1-form on .

*Definition 1. * is a lightlike hypersurface of a semi-Riemannian manifold . A point is said to be umbilical if
where is the local second fundamental form of defined around and . One says is totally umbilical if any point of is umbilical.

Then it is easy to see that is totally umbilical if and only if on there exists a smooth function on such that for all vector fields .

*Definition 2. *Let be a lightlike hypersurface; then the screen bundle is totally umbilical if on any coordinate neighbourhood there exist a smooth function such that
In case () on one says is totally geodesic (properly totally umbilical).

#### 3. The Shape Operator and Newton Transformation

In this section we will introduce the shape operators and the corresponding Newton transformations .

From [10] we have the relationship between two local second fundamental forms with the shape operators where , , and . From (11) we have that belong to screen bundle , if and is an integrable distribution; then it is easy to know is self-adjoint linear operator in each tangent space ; its eigenvalues and are the principal curvature of the lightlike hypersurface. If are the eigenvector of corresponding to the eigenvalue and on , respectively, where , and , then the matrix of with respect to the orthogonal basis is given by Now we restrict on ; then it can be denoted by

Proposition 3. *Let be a lightlike hypersurface of ; then the screen bundle is totally umbilical if and only if there exist a smooth function such that
*

Proposition 4 (see [10]). * is a lightlike hypersurface of a semi-Riemannian manifold of constant sectional curvature and totally umbilical screen bundle ; that is, there exists a smooth function making and . Then is totally umbilical and immersed into if . Under this condition, we say is totally geodesic immersed into if and only if is a solution of the partial differential equation .*

The proof of Proposition 4 has been given by Duggal and Bejancu in [10].

From Proposition 4 and the characters of totally umbilical lightlike hypersurface, we can obtain the following corollary.

Corollary 5. *Let be a lightlike hypersurface of , where the sectional curvature of ambient space is constant . Now one supposes the screen bundle is integrable; and , where is a smooth function. Then is totally umbilical; that is, , where . Besides , , and satisfied the following equation:
*

From Corollary 5 we know that is a totally umbilical hypersurface in ; if there exists a smooth function such that on screen bundle and , where denotes the identity in .

Associated with the shape operator there are algebraic invariants given by as follows: where are the elementary symmetric functions in given by Then the characteristic polynomial of restrict on can be denoted by where . Now we denote by ; is the -mean curvature of the screen bundle with respect to the shape operator . In particular, when , is the main extrinsic mean curvature of the screen bundle with respect to ; when , defines a 2-mean curvature which is intrinsic. Thus, if all are nonzero, we can choose a proper orientation to make and unchanged.

The curvature tensor of the ambient space can be written as follows: where is the tangential component of .

Now we denote to be the basis on and the Ricci tensor of , where ; that is, , the basis of ; then from [10] we can obtain that So we obtain where , , is the 1-form induced by . When we have By using the equations above we obtain that when the Ricci tensor is symmetric; then we have the following proposition.

Proposition 6. *Let be a lightlike hypersurface of a RW spacetime which is equipped with a conformal vector field . If the Ricci tensor of the induced connection is symmetric, then there exists a pair on such that the corresponding 1-form vanishes on any ; besides and are all nonzero.*

*Proof. *It is already known that there exist vector fields and such that , .

Since the Ricci tensor of is symmetric, then we have that the -form induced by is closed; that is, ; so we have .

If , then the theorem have been proved.

If , we take , , the corresponding -form is , where
By using Poincaré lemma we obtain , where . Then we can let , so for any . Since , , then the is the vector fields satisfying the proposition.

From Proposition 6 we obtain that if the Ricci tensor of is symmetric, then we have where , is the covariant derivative of .

Because the ambient space is RW spacetime, the sectional curvature is constant; then when , we have that has no component in . Now (24) becomes

Now we introduce the Newton transformations which arises from the shape operator . According to the definition of the -mean curvature of screen bundle , the Newton transformations are given by where denotes the identify in , and . Besides, According to the Cayley-Hamilton theorem, we have . When is even, the definition of does not depend on the chosen Gauss map; when is odd there is a change of sign in the definition of .

It is easy to know that is also a self-adjoint linear operator with respect to ; besides, and can be a simultaneous diagonalization. If is the orthogonal frame on which diagonalizes , , are all self-adjoint operators with respect to , then we have where Then we obtain where .

The divergence of on screen bundle is defined by

Proposition 7. *Let be a lightlike hypersurface of a RW spacetime in which the screen bundle is integrable and Ricci tensor of the induced connection is symmetric; when the sectional curvature of the ambient spacetime is constant, then the divergence of Newton transformations is given by the following inductive formula:
**
where denotes the screen bundle component of . Besides, for any vector field it satisfies
*

*Proof. *It is easy to know , when , ; we have
Then
Using (24) we get for any vector field ; we have
By using equation (4.4) in [7], we have
From (24) and the above formula, we have

Since the sectional curvature of the RW spacetime is constant, then it is easy to know , for all vector fields . From (25) and (34) we have , ; thus we have the corollary below.

Corollary 8. *Let be a lightlike hypersurface of a conformally stationary spacetime in which the screen bundle is integrable and Ricci tensor of the induced connection is symmetric; then we have that the Newton transformations on are divergence-free: , .*

#### 4. Minkowski Integral Formula for Lightlike Hypersurface in RW Spacetime

In this section, by using divergence theorem we derive some general integral formulae for compact lightlike hypersurface in a RW spacetime . In order to get that, we consider the vector field , ; it determines a nonvanishing future-pointing closed conformal vector field on ; besides, for any vector field tangent to at a point .

We denote the height function of by , which is the restriction of the projection to ; is given by . Observe the gradient of on is given by then we can get the gradient of on is where , is the tangential component of . Now we denote is an arbitrary primitive of ; that is, ; so we can consider as a reparametrization of the height function. Then we have where . From (41) we get for any .

Since for every , then we have From we have

If we take as the tangential component of along the hypersurface, we have where and ; then we have

Now we have

In order to compute the divergence of , we have to expend the definition of operator to the tangential bundle. Now let and be the local orthogonal frame on ; then with respect to the basis can be denoted by in this case, we denote it by to distinguish it from restricted on . By a direct calculation we obtain

Then , , . From all the above, we have the theorem below.

Theorem 9. *Let be a compact lightlike hypersurface immersed into a RW spacetime , where the screen bundle is integrable, , and the Ricci tensor of induced connection is symmetric. Then the following formula holds:
*

*Proof. *Since the sectional curvature of RW spacetime is constant, then we have the calculation about the divergence of as follows:
Since the sectional curvature of RW spacetime is constant, by using Proposition 7 we have ; then
From divergence theorem we have

We call the formula (53) as Minkowski-type integral formula.

#### 5. Umbility of the Compact Lightlike Hypersurface in RW Spacetime

In this section, we will derive some application of Minkowski formulae for the case of lightlike hypersurface immersed into a spatially closed RW spacetime, where the screen bundle of the hypersurface has constant -mean curvature; under this condition by using (35), it is easy to know that if is totally umbilical, then must be constant.

Theorem 10. *Let be spatially closed RW spacetime; then every compact lightlike hypersurface immersed into , where , screen bundle is integrable and Ricci tensor of the induced connection is symmetric. If of the screen bundle with respect to is constant, then the hypersurface must be totally umbilical.*

*Proof. *Using the Theorem 9, It is easy to see that if is constant, we can consider , then multiplying the Minkowski-type formula by we have
When , we obtain that the Minkowski formula is
Subtracting the two formulas above we get
from the Cauchy-Schwarz inequality we have
where the equality holds only at the point . From we have that the equality holds; then there exists constant such that on screen bundle. Besides, ; then by using Corollary 5 we have that the hypersurface is totally umbilical.

Theorem 11. *Let be a spatially closed RW spacetime. Assume that is a compact lightlike hypersurface immersed into with ; screen bundle is integrable and the Ricci tensor of induced connection is symmetric. If is a positive constant, then is a totally umbilical lightlike hypersurface of .*

*Proof. *Using the Theorem 9 and multiplying the formula (53) by we have
When , we obtain that the Minkowski formula (53) is
Subtracting the two formulas above we get
Since , from (60) we have , so does not vanish on . Since is an intrinsic quantity and it does not depend on the chosen orientation, then we can choose an appropriate orientation such that . From the generalization Cauchy-Schwarz inequality in [17] we have ; then
where the equality holds only at the point when . Besides, , so we have ; then it is easy to know that there exists constant making that on screen bundle. Since , then from Corollary 5 we have that is totally umbilical.

Theorem 12. *Let be a spatially closed RW spacetime. Assume that is a compact lightlike hypersurface immersed into with ; screen bundle is integrable and the Ricci tensor of induced connection is symmetric. If and are both constants , then the hypersurface must be totally umbilical.*

*Proof. *Using Theorem 9 and multiplying the formula (53) by constant , then we obtain
Multiplying the -th formula by constant , then we have
Subtracting the two formulas above we get
By using the generalization of Cauchy-Schwarz inequality we have
where the equality holds only at the point . Besides, , so we have ; then there exists constant making that on screen bundle. Since ; thus by using the Corollary 5, we have that is totally umbilical.

Theorem 13. *Let be a spatially closed RW spacetime. Assume that is a compact lightlike hypersurface immersed into with ; screen bundle is integrable and the Ricci tensor of induced connection is symmetric. If and is constant, where , then the hypersurface is totally umbilical in .*

*Proof. *In the same way as the proof of Theorem 11, we have
From the generalization of Cauchy-Schwarz inequality we have for any
where the equality holds only at the point . Since , , then we have
where the equality holds only at the point . From we have ; then there exists constant making that on screen bundle. Since , thus by using the Corollary 5, we have that is totally umbilical.

#### 6. Compact Lightlike Hypersurface with Elliptic Point

In this section, an elliptic point we mean is a point where all the principal curvature of restricted on the screen bundle has the same sign. Now we will derive the result about the existence of elliptic point lightlike hypersurface.

Lemma 14. *Let be a compact lightlike hypersurface immersed into a spatially closed RW spacetime, where , screen bundle is integrable and the Ricci tensor of induced connection is symmetric. When does not vanish on , then (i). If on , then there exists an elliptic point of with respect to its future-pointing Gauss map. (ii) If on , then there exists an elliptic point of with respect to its past-pointing Gauss map.*

*Proof. *When , let us choose on the future-pointing Gauss map and let be the point where the height function attains its minimum on , so we have , and . From we have
where is a basis of principal direction at , , so we have
then we obtain the result.

In a similar way, when we choose the past-pointing Gauss map on such that ; consider to be the point where the height function attains its maximum on . Now we have ,
From we have
Then we finish the proof.

Theorem 15. *Let () be a spatially closed RW spacetime. Assume that is a compact lightlike hypersurface immersed into which contains an elliptic point, ; screen bundle is integrable and the Ricci tensor of induced connection is symmetric. If is constant, where , then the hypersurface is totally umbilical in .*

*Proof. *From Lemma 14 we have that there exists elliptic point with respect to the future-pointing Gauss map , such that all the principle curvature of restricts on ; then constant is positive. By using Garding inequality we have
where the equality holds only at the point where . From on we have that . Integrating this inequality and using the Minkowski formula we obtain
So we have
Besides, we already know that and , so we have ; then ; that is, there exists constant making that on screen bundle. Since , thus by using the Corollary 5 we know that is totally umbilical.

When the hypersurface immersed into the RW spacetime with linearly related -mean curvature, then we have the following totally umbilical result.

Theorem 16. *Let () be a spatially closed RW spacetime. Assume that is a compact lightlike hypersurface immersed into with ; screen bundle is integrable and the Ricci tensor of induced connection is symmetric. If , there are exists integers and , where or and the higher-order mean curvature restricted on is linearly related by
**
where are nonnegative; then the hypersurface is totally umbilical.*

*Proof. *(i) When
from the formula (79) and the Minkowski formula, we have
then we have
Since , we suppose that is positive on ; then from Lemma 14 we know that there exists a point where all the principal curvature of restricted on is negative. Denote as the connected component of containing the elliptic point , so is a nonempty subset of . If is also closed, then ; now we will prove that it is closed. From we have that there exists at least one positive coefficient , such that ; then by using Garding inequalities we have for any point ,
where , ; then we obtain .

If , then on , so is closed.

If , then on we have , so on ; that is, ; then also is closed.

Thus and (82) always holds on .

From formula (83) and Cauchy-Schwarz inequality we have and
where the equality holds only at the point , . Then it equals
where the equality holds only at the point , .

From (79) and (85) we get that
then we obtain . By using and (82) we have
It means that . Since then by using Corollary 5 we have that is totally umbilical.

(ii) When , now we just need to prove the case and ; others are the same with (i); by using the Minkowski formula we have that
so we have
In a similar way, we assume is positive on , then there exists a point where all the principal curvatures of restricted on screen bundle are negative. In the same way we obtain that and at each point . From Newton inequality we have
where the equality holds only at the points . From (79) we have
then from (89) we obtain that
where the equality holds only at the points . By using Corollary 5 we have that is totally umbilical.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work is supported by NSFC (no. 11371076).