# Totally Umbilical Lightlike Hypersurfaces in Robertson-Walker Spacetimes

Authors: Junhong Dong; Ximin Liu
Journal: ISRN Geometry
 (2014)

---

## 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.


---

## Body

## 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 (
M
,
g
,
S
(
T
M
)
) be a compact lightlike hypersurface of a conformally stationary spacetime M
¯ of constant sectional curvature, on which the screen bundle S
(
T
M
) is integrable and Ricci tensor of the induced connection ∇ is symmetric; then the following conclusion holds:
(1)
∫
M
‍
(
ϕ
H
r
+
〈
∂
t
,
ξ
〉
H
r
+
1
′
+
〈
∂
t
,
N
〉
H
r
+
1
)
d
V
=
0
.The Minkowski Formula II. Let (
M
,
g
,
S
(
T
M
)
) be a compact lightlike hypersurface immersed into a conformally stationary spacetime (
M
¯
,
g
) of constant sectional curvature, where the screen bundle S
(
T
M
) is integrable and the Ricci tensor of induced connection ∇ is symmetric; then
(2)
∫
M
‍
(
ϕ
H
r
′
+
〈
∂
t
,
ξ
〉
H
r
+
1
′
)
d
V
=
0
.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. Theorems10–13).This paper is organized as follows. In Section2, 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

LetF
n
+
1 be an (
n
+
1
)-dimensional (
n
≥
1
) Riemannian manifold and let I be a 1-dimensional manifold (either a circle or an open interval of R). We denote M
¯
n
+
2 by the (
n
+
2
)-dimensional product manifold endowed with the Lorentzian metric
(3)
〈
,
〉
=
-
d
t
2
+
f
2
(
t
)
〈
,
〉
F
,
where f
>
0 is a positive smooth function on I, and 〈
,
〉
F is the Riemannian metric on F
n
+
1; then we have that M
¯ is a Lorentzian warped product with Lorentzian base (
I
,
-
d
t
2
), Riemannian fiber F
n
+
1, 〈
,
〉
F, and warping function f, when the sectional curvature of Riemannian factor F
n
+
1 is constant; then we call -
I
×
F
n
+
1 as Robertson-Walker (RW) spacetime.Consider smooth immersionψ
:
M
n
+
1
↪
-
I
×
F
n
+
1 of an n
+
1-dimensional connected lightlike manifold M into a RW spacetime, where the induced metric on M is g; then we have that M
n
+
1 is a lightlike hypersurface.Let∂
t
=
(
∂
/
∂
t
)
(
t
,
x
), (
t
,
x
)
∈
-
I
×
F
n
+
1 be the unitary timelike conformal vector field globally defined on the RW spacetime M
¯; then consider lightlike hypersurface (
M
,
g
,
S
(
T
M
)
); there exists a pair lightlike vector fields {
N
,
ξ
} defined on M which satisfied the following formulas:
(4)
〈
ξ
,
ξ
〉
=
〈
N
,
N
〉
=
0
,
〈
ξ
,
N
〉
=
1
,
(5)
〈
∂
t
,
ξ
〉
≠
0
,
〈
∂
t
,
N
〉
≠
0
.
There exists a unitary timelike vector field T
=
N
-
ξ defined on M which is the same time-orientation as ∂
t; we will refer to T as the future-pointing Gauss map of the hypersurface; its opposite is the past-pointing Gauss map, where we can suppose 〈
ξ
,
∂
t
〉
>
0 on M in the future-pointing.Now we denote∇
,
∇
* to be the connection of M and S
(
T
M
), respectively, P the projection morphism of T
M onto S
(
T
M
); then the Gauss and Weingarten formula for the hypersurface in M
¯ are given as follows:
(6)
∇
¯
X
Y
=
∇
X
Y
+
h
(
X
,
Y
)
=
∇
X
Y
+
B
(
X
,
Y
)
N
,
∇
¯
X
N
=
-
A
N
X
+
∇
X
t
N
=
-
A
N
X
+
τ
(
X
)
N
,
∇
X
P
Y
=
∇
X
*
P
Y
+
h
*
(
X
,
P
Y
)
=
∇
X
*
P
Y
+
C
(
X
,
P
Y
)
ξ
,
∇
X
ξ
=
-
A
ξ
*
X
-
τ
(
X
)
ξ
,
A
ξ
*
ξ
=
0
,
where X
,
Y
∈
Γ
(
T
M
), ξ
∈
Γ
(
Rad
T
M
), N
∈
Γ
(
tr
⁡
(
T
M
)
), h and h
* are the second fundamental forms of M and S
(
T
M
), B and C are the local second fundamental forms on Γ
(
T
M
) and Γ
(
S
(
T
M
)
), respectively, A
N and A
ξ
* are the local shape operator on M and S
(
T
M
), respectively, and τ is the 1-form on T
M.Definition 1.
(
M
,
g
,
S
(
T
M
)
) is a lightlike hypersurface of a semi-Riemannian manifold (
M
¯
,
g
¯
). A point u
∈
M is said to be umbilical if
(7)
B
u
(
X
u
,
Y
u
)
=
k
g
u
(
X
u
,
Y
u
)
,
X
u
,
Y
u
∈
T
u
M
,
where B is the local second fundamental form of M defined around u and k
∈
R. One says M is totally umbilical if any point of M is umbilical.Then it is easy to see thatM is totally umbilical if and only if on M there exists a smooth function ρ on M such that
(8)
B
(
X
,
Y
)
=
ρ
g
(
X
,
Y
)
,
for all vector fields X
,
Y
∈
Γ
(
T
M
).Definition 2.
Let(
M
,
g
,
S
(
T
M
)
) be a lightlike hypersurface; then the screen bundle is totally umbilical if on any coordinate neighbourhood U
⊂
M there exist a smooth function λ such that
(9)
C
(
X
,
P
Y
)
=
λ
g
(
X
,
P
Y
)
,
X
,
Y
∈
Γ
(
T
M
)
.
In case λ
=
0 (λ
≠
0) on U one says S
(
T
M
) 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 transformationsT
r.From [10] we have the relationship between two local second fundamental forms with the shape operators
(10)
B
(
X
,
Y
)
=
〈
A
ξ
*
X
,
Y
〉
,
C
(
X
,
P
Y
)
=
〈
A
N
X
,
P
Y
〉
,
(11)
B
(
X
,
ξ
)
=
〈
A
ξ
*
X
,
N
〉
=
0
,
〈
A
N
X
,
N
〉
=
0
,
where X
,
Y
∈
Γ
(
T
M
), ξ
∈
Γ
(
T
M
⊥
), and N
∈
Γ
(
tr
⁡
T
M
). From (11) we have that A
N
X belong to screen bundle S
(
T
M
), if A
N
ξ
=
0 and S
(
T
M
) is an integrable distribution; then it is easy to know A
N is self-adjoint linear operator in each tangent space T
p
M; its eigenvalues k
1
,
k
2
,
…
,
k
n and k
n
+
1 are the principal curvature of the lightlike hypersurface. If {
e
1
,
…
,
e
n
,
ξ
} are the eigenvector of A
N corresponding to the eigenvalue k
1
(
p
)
,
k
2
(
p
)
,
…
,
k
n
(
p
) and k
n
+
1
(
p
) on T
p
M, respectively, where A
N
(
e
i
)
p
=
k
i
(
p
)
e
i, 1
≤
i
≤
n and k
n
+
1
=
0, then the matrix of A
N with respect to the orthogonal basis {
e
1
,
…
,
e
n
,
ξ
} is given by
(12)
(
k
1
⋯
0
0
⋮
⋱
⋮
⋮
0
⋯
k
n
0
0
⋯
0
0
)
.
Now we restrict A
N on S
(
T
M
); then it can be denoted by
(13)
(
k
1
⋯
0
⋮
⋱
⋮
0
⋯
k
n
)
.Proposition 3.
Let(
M
,
g
,
S
(
T
M
)
) be a lightlike hypersurface of (
M
¯
,
g
¯
); then the screen bundle is totally umbilical if and only if there exist a smooth function λ such that
(14)
A
N
X
=
λ
X
,
A
N
ξ
=
0
,
X
∈
Γ
(
S
(
T
M
)
)
.Proposition 4 (see [10]).
(
M
n
+
1
,
g
,
S
(
T
M
)
) is a lightlike hypersurface of a semi-Riemannian manifold (
M
¯
,
g
¯
) of constant sectional curvature c and totally umbilical screen bundle S
(
T
M
); that is, there exists a smooth function λ making A
N
X
=
λ
X and A
N
ξ
=
0. Then M is totally umbilical and immersed into M
¯ if λ
≠
0. Under this condition, we say M is totally geodesic immersed into M
¯ if and only if λ is a solution of the partial differential equation ξ
(
λ
)
-
λ
τ
(
ξ
)
+
c
=
0.The proof of Proposition4 has been given by Duggal and Bejancu in [10].From Proposition4 and the characters of totally umbilical lightlike hypersurface, we can obtain the following corollary.Corollary 5.
Let(
M
,
g
,
S
(
T
M
)
) be a lightlike hypersurface of (
M
¯
(
c
)
,
g
¯
), where the sectional curvature of ambient space M
¯ is constant c. Now one supposes the screen bundle S
(
T
M
) is integrable; A
N
ξ
=
0 and A
N
X
=
λ
X
(
λ
≠
0
), where λ is a smooth function. Then M is totally umbilical; that is, A
ξ
Y
=
ρ
Y, where X
,
Y
∈
Γ
(
S
(
T
M
)
). Besides ρ, λ, and c satisfied the following equation:
(15)
ξ
(
λ
)
-
λ
τ
(
ξ
)
+
c
=
ρ
λ
,
ξ
∈
Γ
(
T
M
⊥
)
.From Corollary5 we know that M
n
+
1 is a totally umbilical hypersurface in M
¯
(
c
); if there exists a smooth function λ such that A
N
=
λ
I on screen bundle and A
N
ξ
=
0, where I denotes the identity in S
(
T
M
).Associated with the shape operatorA
N there are n algebraic invariants given by S
r as follows:
(16)
S
r
(
p
)
=
σ
r
(
k
1
(
p
)
,
k
2
(
p
)
,
…
,
k
n
(
p
)
)
,
1
≤
r
≤
n
,
where σ
:
R
n
→
R are the elementary symmetric functions in R
n given by
(17)
σ
r
(
x
1
,
x
2
,
…
,
x
n
)
=
∑
i
1
<
⋯
<
i
r
‍
x
i
1
⋯
x
i
r
.
Then the characteristic polynomial of A
N restrict on S
(
T
M
) can be denoted by
(18)
det
⁡
(
t
I
-
A
N
)
=
∑
r
=
0
n
(
-
1
)
r
S
r
t
n
-
r
,
where S
0
=
1. Now we denote H
r by (
n
r
)
H
r
=
(
-
1
)
r
S
r; H
r is the r-mean curvature of the screen bundle with respect to the shape operator A
N. In particular, when r
=
1, H
1
=
(
1
/
n
)
tr
⁡
(
A
N
) is the main extrinsic mean curvature of the screen bundle with respect to A
N; when r
=
2, H
2 defines a 2-mean curvature which is intrinsic. Thus, if all k
i are nonzero, we can choose a proper orientation to make H
1
>
0 and H
2 unchanged.The curvature tensorR
¯ of the ambient space M
¯ can be written as follows:
(19)
R
¯
(
X
,
Y
)
Z
=
R
(
X
,
Y
)
Z
+
A
h
(
X
,
Z
)
Y
-
A
h
(
Y
,
Z
)
X
+
(
∇
X
h
)
(
Y
,
Z
)
-
(
∇
Y
h
)
(
X
,
Z
)
,
R
(
X
,
Y
)
Z
=
(
R
¯
(
X
,
Y
)
Z
)
T
-
A
h
(
X
,
Z
)
Y
+
A
h
(
Y
,
Z
)
X
,
where (
∇
X
h
)
(
Y
,
Z
)
=
∇
X
t
(
h
(
Y
,
Z
)
)
-
h
(
∇
X
Y
,
Z
)
-
h
(
Y
,
∇
X
Z
)
,
(
R
¯
(
X
,
Y
)
Z
)
T is the tangential component of R
¯
(
X
,
Y
)
Z.Now we denote{
E
0
,
E
i
,
N
} to be the basis on M and Ric the Ricci tensor of M, where 1
≤
i
≤
n; that is, ξ
=
E
0, {
E
i
} the basis of S
(
T
M
); then from [10] we can obtain that
(20)
Ric
(
E
i
,
E
j
)
=
Ric
¯
(
E
i
,
E
j
)
-
g
¯
(
R
¯
(
E
i
,
E
0
)
E
j
,
N
)
-
g
k
t
(
C
(
E
k
,
E
t
)
B
(
E
i
,
E
j
)
-
C
(
E
i
,
E
t
)
B
(
E
j
,
E
k
)
)
.
So we obtain
(21)
Ric
(
E
i
,
E
j
)
-
Ric
(
E
j
,
E
i
)
=
g
k
t
{
C
(
E
i
,
E
t
)
B
(
E
j
,
E
k
)
-
C
(
E
j
,
E
t
)
B
(
E
i
,
E
k
)
}
+
g
¯
(
R
¯
(
E
j
,
E
i
)
E
0
,
N
)
=
A
j
k
B
k
i
-
A
i
k
B
k
j
+
R
¯
0
i
j
0
=
2
d
τ
(
∂
∂
u
k
,
∂
∂
u
h
)
,
where 1
≤
k, t
≤
n, τ is the 1-form induced by S
(
T
M
). When i
=
0 we have
(22)
Ric
(
E
0
,
E
j
)
-
Ric
(
E
j
,
E
0
)
=
-
A
0
k
B
k
j
+
R
¯
00
k
0
=
2
d
τ
(
E
0
,
E
j
)
.
By using the equations above we obtain that d
τ
=
0 when the Ricci tensor Ric is symmetric; then we have the following proposition.Proposition 6.
Let(
M
,
g
,
S
(
T
M
)
) be a lightlike hypersurface of a RW spacetime which is equipped with a conformal vector field ∂
t. If the Ricci tensor of the induced connection ∇ is symmetric, then there exists a pair {
ξ
,
N
} on U such that the corresponding 1-form τ vanishes on any U
⊂
M; besides 〈
∂
t
,
ξ
〉 and 〈
∂
t
,
N
〉 are all nonzero.Proof.
It is already known that there exist vector fieldsξ and N such that 〈
∂
t
,
ξ
〉
≠
0, 〈
∂
t
,
N
〉
≠
0.
Since the Ricci tensor of∇ is symmetric, then we have that the 1-form τ induced by S
(
T
M
) is closed; that is, d
τ
≡
0; so we have τ
=
α.
Ifτ
=
α
=
0, then the theorem have been proved.
Ifτ
=
α
≠
0, we take ξ
¯
=
α
ξ, N
¯
=
(
1
/
α
)
N, the corresponding 1-form is τ
¯, where
(23)
τ
¯
(
X
)
=
g
¯
(
∇
X
t
N
¯
,
ξ
¯
)
=
α
X
(
1
α
)
+
τ
(
X
)
,
τ
(
X
)
=
τ
¯
(
X
)
-
α
X
(
1
α
)
=
τ
¯
(
X
)
-
X
(
log
⁡
α
)
.
By using Poincaré lemma we obtain τ
(
X
)
=
X
(
f
), where f
∈
C
∞
(
U
). Then we can let α
=
e
f, so τ
¯
(
X
)
=
0 for any X
∈
Γ
(
T
M
U
). Since 〈
∂
t
,
ξ
¯
〉
=
α
〈
∂
t
,
ξ
〉
≠
0, 〈
∂
t
,
N
¯
〉
=
(
1
/
α
)
〈
∂
t
,
N
〉
≠
0, then the {
ξ
¯
,
N
¯
} is the vector fields satisfying the proposition.From Proposition6 we obtain that if the Ricci tensor of ∇ is symmetric, then we have
(24)
〈
R
¯
(
X
,
Y
)
P
Z
,
N
〉
=
〈
(
∇
X
C
)
(
Y
,
P
Z
)
-
(
∇
Y
C
)
(
X
,
P
Z
)
,
ξ
〉
=
〈
(
∇
X
A
N
)
Y
-
(
∇
Y
A
N
)
X
,
P
Z
〉
,
where (
∇
X
C
)
(
Y
,
P
Z
)
=
X
(
C
(
Y
,
P
Z
)
)
-
C
(
∇
X
Y
,
P
Z
)
-
C
(
Y
,
∇
X
*
P
Z
), ∇
X
A
N is the covariant derivative of A
N.Because the ambient space is RW spacetime, the sectional curvature is constant; then whenX
,
Y
∈
Γ
(
S
(
T
M
)
), we have that R
¯
(
X
,
Y
)
Z
=
c
{
g
(
Y
,
Z
)
X
-
g
(
X
,
Z
)
Y
} has no component in Γ
(
T
M
⊥
). Now (24) becomes
(25)
(
∇
X
A
N
)
Y
=
(
∇
Y
A
N
)
X
.Now we introduce the Newton transformations(26)
T
r
:
Γ
(
S
(
T
M
)
)
⟶
Γ
(
S
(
T
M
)
)
,
0
≤
r
≤
n
,
which arises from the shape operator A
N. According to the definition of the r-mean curvature of screen bundle S
(
T
M
), the Newton transformations are given by
(27)
T
r
=
(
n
r
)
H
r
I
+
(
n
r
-
1
)
H
r
-
1
A
N
+
⋯
+
(
n
1
)
H
1
A
N
r
-
1
+
A
N
r
,
where I denotes the identify in Γ
(
S
(
T
M
)
), 1
≤
r
≤
n and T
0
=
I. Besides,
(28)
T
r
=
(
n
r
)
H
r
I
+
A
N
T
r
-
1
.
According to the Cayley-Hamilton theorem, we have T
n
=
0. When r is even, the definition of T
r does not depend on the chosen Gauss map; when k is odd there is a change of sign in the definition of T
r.It is easy to know thatT
r
(
p
) is also a self-adjoint linear operator with respect to A
N
(
p
); besides, A
N
(
p
) and T
r
(
p
) can be a simultaneous diagonalization. If {
e
1
,
…
,
e
n
} is the orthogonal frame on S
(
T
M
) which diagonalizes A
N
p, A
N
p
(
e
i
)
=
k
i
(
p
)
e
i, T
r are all self-adjoint operators with respect to A
N, then we have
(29)
T
r
(
e
i
)
=
λ
i
,
r
(
e
i
)
,
where
(30)
λ
i
,
r
=
∑
i
1
<
⋯
<
i
r
,
i
j
≠
i
‍
k
i
1
⋯
k
i
r
.
Then we obtain
(31)
tr
⁡
(
T
r
)
=
c
r
H
r
,
tr
⁡
(
A
N
∘
T
r
)
=
-
c
r
H
r
+
1
,
where c
r
=
(
n
-
r
)
(
n
r
).The divergence ofT
r on screen bundle is defined by
(32)
di
v
S
(
T
M
)
T
r
=
tr
⁡
(
∇
T
r
)
=
∑
i
=
1
n
‍
(
∇
e
i
T
r
,
e
i
)
.Proposition 7.
Let(
M
,
g
,
S
(
T
M
)
) be a lightlike hypersurface of a RW spacetime in which the screen bundle S
(
T
M
) 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 T
r is given by the following inductive formula:
(33)
di
v
S
(
T
M
)
T
0
=
0
,
(34)
di
v
S
(
T
M
)
T
r
=
A
N
(
div
⁡
T
r
-
1
)
+
∑
i
=
1
n
‍
(
R
¯
(
N
,
T
r
-
1
e
i
)
e
i
)
*
,
where (
R
¯
(
N
,
X
)
Y
)
* denotes the screen bundle component of R
¯
(
N
,
X
)
Y. Besides, for any vector field X
∈
Γ
(
S
(
T
M
)
) it satisfies
(35)
〈
div
S
(
T
M
)
T
r
,
X
〉
=
∑
j
=
1
r
∑
i
=
1
n
(
-
1
)
j
〈
R
¯
(
ξ
,
T
r
-
j
e
i
)
e
i
,
A
ξ
j
-
1
X
〉
.Proof.
It is easy to knowdi
v
S
(
T
M
)
T
0
=
div
⁡
I
=
0, when r
≥
1, X
,
Y
∈
Γ
(
S
(
T
M
)
); we have
(36)
(
∇
X
T
r
)
Y
=
(
n
r
)
〈
∇
H
r
,
X
〉
Y
+
∇
X
(
A
N
T
r
-
1
)
Y
=
(
n
r
)
〈
∇
H
r
,
X
〉
Y
+
(
∇
X
A
N
)
T
r
-
1
Y
+
A
N
(
(
∇
X
T
r
-
1
)
Y
)
.
Then
(37)
di
v
S
(
T
M
)
T
r
=
∑
i
=
1
n
‍
(
∇
e
i
T
r
)
(
e
i
)
=
(
n
r
)
∇
H
r
+
∑
i
=
1
n
‍
(
∇
e
i
A
N
)
(
T
r
-
1
e
i
)
+
A
N
(
di
v
S
(
T
M
)
T
r
-
1
)
.
Using (24) we get for any vector field X
∈
Γ
(
S
(
T
M
)
); we have
(38)
〈
(
∇
e
i
A
N
)
(
T
r
-
1
e
i
)
,
X
〉
=
〈
(
∇
X
A
N
)
e
i
,
T
r
-
1
e
i
〉
+
〈
R
¯
(
e
i
,
X
)
T
r
-
1
e
i
,
N
〉
=
〈
T
r
-
1
(
(
∇
e
i
A
N
)
e
i
)
,
e
i
〉
-
〈
R
¯
(
N
,
T
r
-
1
e
i
)
e
i
,
X
〉
.
〈
di
v
S
(
T
M
)
T
r
,
X
〉
=
(
n
r
)
〈
∇
H
r
,
X
〉
+
tr
⁡
(
T
r
-
1
∇
X
A
N
)
-
∑
i
=
1
n
‍
〈
R
¯
(
N
,
T
r
-
1
e
i
)
e
i
,
X
〉
+
〈
A
N
(
di
v
S
(
T
M
)
T
r
-
1
)
,
X
〉
.
By using equation (4.4) in [7], we have
(39)
tr
⁡
(
T
r
-
1
∇
X
A
N
)
=
(
n
r
)
〈
∇
H
r
,
X
〉
.
From (24) and the above formula, we have
(40)
〈
di
v
S
(
T
M
)
T
r
,
X
〉
=
-
〈
R
¯
(
N
,
T
r
-
1
e
i
)
e
i
,
X
〉
+
〈
A
N
(
di
v
S
(
T
M
)
T
r
-
1
)
,
X
〉
=
〈
-
R
¯
(
N
,
T
r
-
1
e
i
)
e
i
+
A
N
(
di
v
S
(
T
M
)
T
r
-
1
)
,
X
〉
=
-
∑
j
=
1
r
‍
∑
i
=
1
n
‍
〈
R
¯
(
N
,
T
r
-
j
e
i
)
e
i
,
A
N
j
-
1
X
〉
.Since the sectional curvature of the RW spacetime is constant, then it is easy to know〈
R
¯
(
N
,
X
)
Y
,
Z
〉
=
〈
R
¯
(
N
,
X
)
Y
,
Z
〉
=
0, for all vector fields X
,
Y
,
Z
∈
Γ
(
S
(
T
M
)
). From (25) and (34) we have di
v
S
(
T
M
)
T
r
=
0, 0
≤
r
≤
n; thus we have the corollary below.Corollary 8.
Let(
M
,
g
,
S
(
T
M
)
) be a lightlike hypersurface of a conformally stationary spacetime in which the screen bundle S
(
T
M
) is integrable and Ricci tensor of the induced connection ∇ is symmetric; then we have that the Newton transformations on S
(
T
M
) are divergence-free: div
⁡
S
(
T
M
)
T
r
=
0, 0
≤
r
≤
n.


## 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 spacetimeM
¯. In order to get that, we consider the vector field K
(
t
,
x
)
=
f
(
t
)
(
∂
/
∂
t
)
(
t
,
x
), (
t
,
x
)
∈
-
I
×
f
F
n
+
1; it determines a nonvanishing future-pointing closed conformal vector field on -
I
×
f
F
n
+
1; besides,
(41)
∇
¯
X
K
=
f
′
(
t
)
X
for any vector field X tangent to -
I
×
f
F
n
+
1 at a point (
t
,
x
).We denote the height function ofM by h, which is the restriction of the projection π
I
(
t
,
x
)
=
t to M; h
:
M
→
I is given by h
=
π
I
∘
ψ. Observe the gradient of π
I on -
I
×
f
F
n
+
1 is given by
(42)
∇
¯
π
I
=
-
〈
∇
¯
π
I
,
∂
t
〉
∂
t
=
-
∂
t
;
then we can get the gradient of h on M is
(43)
∇
h
=
(
∇
¯
π
I
)
T
=
-
∂
t
T
,
where ∂
t
T
=
∂
t
-
〈
ξ
∂
t
,
〉
N, ∂
t
T is the tangential component of ∂
t. Now we denote g
:
I
→
R is an arbitrary primitive of f; that is, g
′
=
f
>
0; so we can consider g as a reparametrization of the height function. Then we have
(44)
∇
g
(
h
)
=
f
(
h
)
∇
h
=
-
f
(
h
)
∂
t
T
=
-
K
T
,
where K
T
=
K
-
〈
K
,
ξ
〉
N. From (41) we get
(45)
∇
¯
X
K
=
f
′
(
h
)
X
,
for any X
∈
Γ
(
T
M
).Since∇
X
K
T
=
f
′
(
h
)
X
+
〈
K
,
ξ
〉
A
N
X for every X
∈
Γ
(
T
M
), then we have
(46)
∇
X
(
∇
h
)
=
-
∇
X
K
T
=
-
f
′
(
h
)
X
-
〈
K
,
ξ
〉
A
N
X
.
From ∇
h
=
(
1
/
f
(
h
)
)
∇
g
(
h
) we have
(47)
∇
X
(
∇
h
)
=
∇
X
(
1
f
(
h
)
∇
g
(
h
)
)
=
-
f
′
(
h
)
f
(
h
)
∇
h
+
1
f
(
h
)
(
-
f
′
(
h
)
X
-
〈
K
,
ξ
〉
A
N
X
)
=
-
(
log
⁡
f
)
′
(
h
)
(
〈
∇
h
,
X
〉
∇
h
+
X
)
-
〈
∂
t
,
ξ
〉
A
N
X
.If we takeK
T as the tangential component of K along the hypersurface, we have
(48)
K
T
=
K
-
〈
K
,
ξ
〉
N
,
where ξ
∈
Γ
(
T
M
⊥
) and 〈
ξ
,
N
〉
=
1; then we have
(49)
∇
X
K
T
=
∇
¯
X
(
K
-
〈
K
,
ξ
〉
N
)
-
B
(
X
,
K
T
)
N
=
f
′
(
h
)
X
-
∇
¯
X
(
〈
K
,
ξ
〉
N
)
-
B
(
X
,
K
T
)
N
=
f
′
(
h
)
X
+
〈
K
,
ξ
〉
A
N
X
+
〈
K
,
A
ξ
*
X
〉
N
-
〈
K
T
,
A
ξ
*
X
〉
N
=
f
′
(
h
)
X
+
〈
K
,
ξ
〉
A
N
X
.Now we have(50)
∇
e
i
K
T
=
〈
f
′
(
h
)
e
i
+
〈
K
,
ξ
〉
A
N
e
i
,
T
r
e
i
〉
=
f
′
(
h
)
〈
e
i
,
T
r
e
i
〉
+
〈
K
,
ξ
〉
〈
A
N
T
r
e
i
,
e
i
〉
.In order to compute the divergence ofK
T, we have to expend the definition of operator T
r to the tangential bundle. Now let A
N
ξ
=
0 and {
e
1
,
…
,
e
n
,
ξ
} be the local orthogonal frame on M; then A
N with respect to the basis can be denoted by
(51)
(
k
1
⋯
0
0
⋮
⋱
⋮
⋮
0
⋯
k
n
0
0
⋯
0
0
)
;
in this case, we denote it by A
¯
N to distinguish it from A
N restricted on S
(
T
M
). By a direct calculation we obtain
(52)
det
⁡
(
t
I
-
A
¯
N
)
=
t
·
det
⁡
(
t
I
-
A
N
)
=
∑
r
=
0
n
‍
(
n
r
)
T
r
t
n
-
r
+
1
.ThenT
¯
r
=
(
T
r
0
0
(
n
r
)
H
r
), di
v
M
T
¯
r
=
di
v
M
(
n
r
)
H
r
·
di
v
M
T
r, T
¯
r
e
i
=
λ
i
,
r
e
i. From all the above, we have the theorem below.Theorem 9.
Let(
M
n
+
1
,
g
,
S
(
T
M
)
) be a compact lightlike hypersurface immersed into a RW spacetime (
M
¯
,
g
), where the screen bundle S
(
T
M
) is integrable, A
N
ξ
=
0, and the Ricci tensor of induced connection ∇ is symmetric. Then the following formula holds:
(53)
∫
M
‍
di
v
M
(
T
r
¯
K
T
)
d
V
=
∫
M
‍
c
r
(
f
′
(
h
)
H
r
-
〈
K
,
ξ
〉
H
r
+
1
)
d
V
=
0
.Proof.
Since the sectional curvature of RW spacetime is constant, then we have the calculation about the divergence ofT
r
¯
K
T as follows:
(54)
di
v
M
(
T
r
¯
K
T
)
=
di
v
S
(
T
M
)
(
T
¯
r
K
T
)
+
〈
ξ
,
ξ
〉
〈
∇
¯
ξ
T
¯
r
K
T
,
ξ
〉
=
di
v
S
(
T
M
)
(
T
¯
r
K
T
)
=
〈
di
v
S
(
T
M
)
T
¯
r
,
K
T
〉
+
∑
i
=
1
n
‍
〈
∇
e
i
K
T
,
T
¯
r
e
i
〉
=
〈
di
v
S
(
T
M
)
T
r
′
·
di
v
S
(
T
M
)
(
n
r
)
H
r
,
K
T
〉
+
f
′
(
h
)
tr
⁡
(
T
r
)
+
〈
K
,
ξ
〉
tr
⁡
(
A
N
T
r
)
=
〈
di
v
S
(
T
M
)
T
r
·
di
v
S
(
T
M
)
(
n
r
)
H
r
,
K
T
〉
+
c
r
(
f
′
(
h
)
H
r
-
〈
K
,
ξ
〉
H
r
+
1
)
.
Since the sectional curvature of RW spacetime is constant, by using Proposition 7 we have di
v
S
(
T
M
)
T
r
=
0; then
(55)
di
v
M
(
T
r
¯
K
T
)
=
c
r
(
f
′
(
h
)
H
r
-
〈
K
,
ξ
〉
H
r
+
1
)
.
From divergence theorem we have
(56)
∫
M
‍
(
f
′
(
h
)
H
r
-
〈
K
,
ξ
〉
H
r
+
1
)
d
V
=
0
.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 constantr-mean curvature; under this condition by using (35), it is easy to know that if M is totally umbilical, then H
r must be constant.Theorem 10.
LetM
¯
=
-
I
×
f
F
n
+
1 be spatially closed RW spacetime; then every compact lightlike hypersurface immersed into M
¯, where A
N
ξ
=
0, screen bundle S
(
T
M
) is integrable and Ricci tensor of the induced connection is symmetric. If H
1 of the screen bundle with respect to A
N is constant, then the hypersurface must be totally umbilical.Proof.
Using the Theorem9, It is easy to see that if H
1
′ is constant, we can consider r
=
0, then multiplying the Minkowski-type formula by H
1 we have
(57)
∫
M
‍
(
f
′
(
h
)
H
1
-
〈
K
,
ξ
〉
H
1
2
)
d
V
=
0
.
When r
=
1, we obtain that the Minkowski formula is
(58)
∫
M
‍
(
f
′
(
h
)
H
1
+
〈
K
,
ξ
〉
H
2
)
d
V
=
0
.
Subtracting the two formulas above we get
(59)
∫
M
‍
〈
K
,
ξ
〉
(
H
1
2
-
H
2
)
d
V
=
0
;
from the Cauchy-Schwarz inequality we have
(60)
H
1
2
-
H
2
=
1
n
(
n
-
1
)
(
∑
i
=
1
n
‍
k
i
2
-
1
n
(
∑
i
=
1
n
‍
k
i
)
2
)
≥
0
,
where the equality holds only at the point k
1
=
k
2
=
⋯
=
k
n. From 〈
K
,
ξ
〉
>
0 we have that the equality holds; then there exists constant λ such that A
N
=
λ
I on screen bundle. Besides, A
N
ξ
=
0; then by using Corollary 5 we have that the hypersurface is totally umbilical.Theorem 11.
LetM
¯
=
-
I
×
f
F
n
+
1 be a spatially closed RW spacetime. Assume that M is a compact lightlike hypersurface immersed into M
¯ with A
N
ξ
=
0; screen bundle S
(
T
M
) is integrable and the Ricci tensor of induced connection ∇ is symmetric. If H
2 is a positive constant, then M is a totally umbilical lightlike hypersurface of M
¯.Proof.
Using the Theorem9 and multiplying the formula (53) by H
2
′ we have
(61)
∫
M
‍
(
f
′
(
h
)
H
2
-
〈
K
,
ξ
〉
H
1
H
2
)
d
V
=
0
.
When r
=
2, we obtain that the Minkowski formula (53) is
(62)
∫
M
‍
(
f
′
(
h
)
H
2
-
〈
K
,
ξ
〉
H
3
)
d
V
=
0
.
Subtracting the two formulas above we get
(63)
∫
M
‍
〈
K
,
ξ
〉
(
H
1
H
2
-
H
3
)
d
V
=
0
Since H
2
>
0, from (60) we have H
1
2
≥
H
2
>
0, so H
1 does not vanish on M. Since H
2 is an intrinsic quantity and it does not depend on the chosen orientation, then we can choose an appropriate orientation such that H
1
>
0. From the generalization Cauchy-Schwarz inequality in [17] we have H
2
2
-
H
1
H
3
≥
0; then
(64)
H
1
H
2
-
H
3
≥
H
1
H
2
-
H
2
2
H
1
=
H
2
H
1
(
H
1
2
-
H
2
)
≥
0
,
where the equality holds only at the point when k
1
′
=
k
2
′
=
⋯
=
k
n
′. Besides, 〈
K
,
ξ
〉
>
0, so we have H
1
H
2
-
H
3
≡
0; then it is easy to know that there exists constant λ making that A
N
=
λ
I on screen bundle. Since A
N
ξ
=
0, then from Corollary 5 we have that M is totally umbilical.Theorem 12.
LetM
¯
=
-
I
×
f
F
n
+
1 be a spatially closed RW spacetime. Assume that M is a compact lightlike hypersurface immersed into M
¯ with A
N
ξ
=
0; screen bundle S
(
T
M
) is integrable and the Ricci tensor of induced connection ∇ is symmetric. If H
r and H
r
-
1 are both constants (
1
≤
r
≤
n
), then the hypersurface M must be totally umbilical.Proof.
Using Theorem9 and multiplying the formula (53) by constant H
r, then we obtain
(65)
∫
M
‍
(
f
′
(
h
)
H
r
-
1
H
r
+
〈
∂
t
,
ξ
〉
H
r
′
2
)
d
V
=
0
.
Multiplying the r-th formula by constant H
r
-
1, then we have
(66)
∫
M
‍
(
f
′
(
h
)
H
r
-
1
H
r
-
〈
K
,
ξ
〉
H
r
-
1
H
r
+
1
)
d
V
=
0
.
Subtracting the two formulas above we get
(67)
∫
M
‍
〈
K
,
ξ
〉
(
H
r
2
-
H
r
-
1
H
r
+
1
)
d
V
=
0
.
By using the generalization of Cauchy-Schwarz inequality we have
(68)
H
r
2
-
H
r
-
1
H
r
+
1
≥
0
,
where the equality holds only at the point k
1
=
k
2
=
⋯
=
k
n. Besides, 〈
K
,
ξ
〉
>
0, so we have H
r
2
-
H
r
-
1
H
r
+
1
≡
0; then there exists constant λ making that A
N
=
λ
I on screen bundle. Since A
N
ξ
=
0; thus by using the Corollary 5, we have that M is totally umbilical.Theorem 13.
LetM
¯
=
-
I
×
f
F
n
+
1 be a spatially closed RW spacetime. Assume that M is a compact lightlike hypersurface immersed into M
¯ with A
N
ξ
=
0; screen bundle S
(
T
M
) is integrable and the Ricci tensor of induced connection ∇ is symmetric. If H
1
>
0
,
…
,
H
r
>
0 and H
r is constant, where 2
≤
r
≤
n
-
1, then the hypersurface M is totally umbilical in M
¯.Proof.
In the same way as the proof of Theorem11, we have
(69)
∫
M
‍
〈
K
,
ξ
〉
(
H
1
H
r
-
H
r
+
1
)
d
V
=
0
,
From the generalization of Cauchy-Schwarz inequality we have for any 1
≤
i
≤
n
(70)
H
i
2
-
H
i
-
1
H
i
+
1
≥
0
,
where the equality holds only at the point k
1
=
k
2
=
⋯
=
k
n. Since H
k
>
0, 1
≤
k
≤
r, then we have
(71)
H
1
≥
H
2
H
1
≥
⋯
≥
H
r
H
r
-
1
,
where the equality holds only at the point k
1
=
k
2
=
⋯
=
k
n. From 〈
K
,
ξ
〉
>
0 we have H
1
H
r
-
H
r
+
1
≡
0; then there exists constant λ making that A
N
=
λ
I on screen bundle. Since A
N
ξ
=
0, thus by using the Corollary 5, we have that M 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 curvaturek
i
(
p
) of A
N 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.
LetM
n
+
1
→
-
I
×
f
F
n
+
1 be a compact lightlike hypersurface immersed into a spatially closed RW spacetime, where A
N
ξ
=
0, screen bundle S
(
T
M
) is integrable and the Ricci tensor of induced connection ∇ is symmetric. When f
′
(
h
) does not vanish on M, then (i). If f
′
(
h
)
>
0 on M, then there exists an elliptic point of M with respect to its future-pointing Gauss map. (ii) If f
′
(
h
)
<
0 on M, then there exists an elliptic point of M with respect to its past-pointing Gauss map.Proof.
Whenf
′
(
h
)
>
0, let us choose on M the future-pointing Gauss map T and let p
0
∈
M be the point where the height function h attains its minimum on M, so we have ∇
h
(
p
0
)
=
0, ∇
2
ρ
(
p
0
)
≥
0 and 〈
ξ
,
∂
t
〉
>
0. From ∇
X
(
∇
h
)
=
-
(
log
⁡
f
)
′
(
h
)
(
〈
∇
h
,
X
〉
∇
h
+
X
)
-
〈
∂
t
,
ξ
〉
A
N
X we have
(72)
∇
2
h
p
0
(
e
i
,
e
i
)
=
-
(
log
⁡
f
)
′
(
h
)
(
p
0
)
-
〈
∂
t
,
ξ
〉
k
i
(
p
0
)
≥
0
,
where {
e
1
,
…
,
e
n
} is a basis of principal direction at p
0, i
=
1
,
…
,
n, so we have
(73)
k
i
(
p
0
)
≤
1
〈
∂
t
,
ξ
〉
(
-
(
log
⁡
f
)
′
(
h
)
(
p
0
)
)
<
0
;
then we obtain the result.
In a similar way, whenf
′
(
h
)
<
0 we choose the past-pointing Gauss map on M such that 〈
∂
t
,
ξ
〉
<
0; consider p
0
∈
M to be the point where the height function h attains its maximum on M. Now we have ∇
h
(
p
0
)
=
0,
(74)
∇
2
h
p
0
(
e
i
,
e
i
)
=
-
(
log
⁡
f
)
′
(
h
)
(
p
0
)
-
〈
∂
t
,
ξ
〉
k
i
(
p
0
)
≤
0
.
From f
′
(
h
)
<
0 we have
(75)
k
i
(
p
0
)
≤
-
1
〈
∂
t
,
ξ
〉
(
log
⁡
f
)
′
(
h
)
(
p
0
)
<
0
.
Then we finish the proof.Theorem 15.
LetM
¯
=
-
I
×
f
F
n
+
1 (n
≥
4) be a spatially closed RW spacetime. Assume that M is a compact lightlike hypersurface immersed into M
¯ which contains an elliptic point, A
N
ξ
=
0; screen bundle S
(
T
M
) is integrable and the Ricci tensor of induced connection ∇ is symmetric. If H
k is constant, where 3
≤
k
≤
n
-
1, then the hypersurface M is totally umbilical in M
¯.Proof.
From Lemma14 we have that there exists elliptic point p
0 with respect to the future-pointing Gauss map T, such that all the principle curvature of A
N restricts on S
(
T
M
); then constant H
k
=
H
k
(
p
0
) is positive. By using Garding inequality we have
(76)
H
1
≥
H
2
1
/
2
≥
⋯
≥
H
k
-
1
1
/
(
k
-
1
)
≥
H
k
1
/
k
>
0
,
where the equality holds only at the point where k
1
=
k
2
=
⋯
=
k
n. From f
′
(
h
)
>
0 on M we have that f
′
(
h
)
H
k
≥
f
′
(
h
)
H
k
(
k
-
1
)
/
k. Integrating this inequality and using the Minkowski formula we obtain
(77)
H
k
∫
M
‍
〈
K
,
ξ
〉
d
V
=
∫
M
‍
f
′
(
h
)
H
k
-
1
d
V
≥
H
k
(
k
-
1
)
/
k
∫
M
‍
f
′
(
h
)
d
V
=
H
k
(
k
-
1
)
/
k
∫
M
‍
〈
K
,
ξ
〉
d
V
.
So we have
(78)
∫
M
‍
(
H
1
-
H
k
1
/
k
)
〈
K
,
ξ
〉
d
V
≤
0
.
Besides, we already know that 〈
K
,
ξ
〉
>
0 and H
1
-
H
k
1
/
k
≥
0, so we have H
1
-
H
k
1
/
k
≡
0; then k
1
=
k
2
=
⋯
=
k
n; that is, there exists constant λ making that A
N
=
λ
I on screen bundle. Since A
N
ξ
=
0, thus by using the Corollary 5 we know that M is totally umbilical.When the hypersurface immersed into the RW spacetime with linearly relatedr-mean curvature, then we have the following totally umbilical result.Theorem 16.
LetM
¯
=
-
I
×
f
F
n
+
1 (n
≥
4) be a spatially closed RW spacetime. Assume that M is a compact lightlike hypersurface immersed into M
¯ with A
N
ξ
=
0; screen bundle S
(
T
M
) is integrable and the Ricci tensor of induced connection ∇ is symmetric. If f
′
(
h
)
≠
0, there are exists integers k and s, where 0
≤
r
<
s
<
n or 0
<
r
<
s
≤
n and the higher-order mean curvature restricted on S
(
T
M
) is linearly related by
(79)
H
s
=
a
k
H
k
+
a
k
+
1
H
k
+
1
+
⋯
+
a
s
H
s
,
where a
k
,
…
,
a
s are nonnegative; then the hypersurface M is totally umbilical.Proof.
(i) When(80)
0
≤
r
<
s
<
n
,
from the formula (79) and the s
+
1
th Minkowski formula, we have
(81)
∫
M
‍
〈
K
,
ξ
〉
H
s
+
1
d
V
=
∫
M
‍
f
′
(
h
)
H
s
d
V
=
∑
i
=
r
s
-
1
‍
a
i
∫
M
‍
f
′
(
h
)
H
i
d
V
=
∑
i
=
r
s
-
1
‍
a
i
∫
M
‍
〈
K
,
ξ
〉
H
i
+
1
d
V
;
then we have
(82)
∫
M
‍
〈
K
,
ξ
〉
(
H
s
+
1
-
∑
i
=
r
s
-
1
‍
a
i
H
i
+
1
)
d
V
=
0
.
Since f
′
(
h
)
≠
0, we suppose that f
′
(
h
) is positive on M; then from Lemma 14 we know that there exists a point p
0
∈
M where all the principal curvature of A
N restricted on S
(
T
M
) is negative. Denote M
0 as the connected component of {
p
∈
M
:
H
s
(
p
)
>
0
} containing the elliptic point p
0, so M
0 is a nonempty subset of M. If M
0 is also closed, then M
0
=
M; now we will prove that it is closed. From H
s
(
p
0
)
>
0 we have that there exists at least one positive coefficient a
k, such that a
k
>
0; then by using Garding inequalities we have for any point p
∈
M
0,
(83)
H
j
s
/
j
≥
H
s
(
p
)
>
0
,
where 1
≤
j
≤
s
-
1, a
j
≥
0; then we obtain H
s
(
p
)
≥
a
k
H
k
(
p
).
Ifk
=
0, then H
s
≥
a
0
>
0 on M
0, so M
0 is closed.
Ifk
≥
1, then on M
0 we have H
k
s
/
k
≥
H
s
≥
a
k
H
k
>
0, so H
k
(
s
-
k
)
/
k
≥
a
k on M
0; that is, H
s
≥
a
k
a
k
k
/
(
s
-
k
)
=
a
k
s
/
(
s
-
k
)
>
0; then M
0 also is closed.
ThusM
0
=
M and (82) always holds on M.
From formula (83) and Cauchy-Schwarz inequality we have H
j
>
0 and
(84)
H
j
2
-
H
j
-
1
H
j
+
1
≥
0
,
where the equality holds only at the point k
1
=
k
2
=
⋯
=
k
n, 1
≤
j
≤
s. Then it equals
(85)
H
s
+
1
H
s
≤
H
s
H
s
-
1
≤
⋯
≤
H
j
+
1
H
j
≤
⋯
≤
H
2
H
1
≤
H
1
,
where the equality holds only at the point k
1
=
k
2
=
⋯
=
k
n, 1
≤
j
≤
s.
From (79) and (85) we get that
(86)
H
s
+
1
H
s
≤
H
s
H
s
-
1
=
∑
j
=
r
s
-
1
‍
a
j
H
j
H
s
-
1
≤
∑
j
=
r
s
-
1
‍
a
j
H
j
+
1
H
s
;
then we obtain H
s
+
1
≤
∑
j
=
r
s
-
1
‍
a
j
H
j
+
1. By using 〈
K
,
ξ
〉
>
0 and (82) we have
(87)
H
s
+
1
=
∑
j
=
r
s
-
1
‍
a
j
H
j
+
1
,
It means that k
1
=
k
2
=
⋯
=
k
n. Since A
N
=
0 then by using Corollary 5 we have that M is totally umbilical.
(ii) When0
<
r
<
s
≤
n, now we just need to prove the case s
=
n and r
>
0; others are the same with (i); by using the Minkowski formula we have that
(88)
∫
M
‍
f
′
(
h
)
H
n
-
1
d
V
=
∑
i
=
r
n
-
1
‍
a
j
∫
M
‍
〈
K
,
ξ
〉
H
j
d
V
=
∑
j
=
r
n
-
1
‍
a
i
∫
M
‍
f
′
(
h
)
H
j
-
1
d
V
,
so we have
(89)
∫
M
‍
f
′
(
h
)
(
H
n
-
1
-
∑
j
=
r
n
-
1
‍
a
i
H
j
-
1
)
d
V
=
0
.
In a similar way, we assume f
′
(
h
) is positive on M, then there exists a point p
0
∈
M where all the principal curvatures of A
N restricted on screen bundle are negative. In the same way we obtain that M
n
=
{
p
∈
M
:
H
n
(
p
)
>
0
}
=
M and H
j
n
/
j
(
p
)
≥
H
n
(
p
)
>
0 at each point p
∈
M. From Newton inequality we have
(90)
H
n
H
n
-
1
≤
H
n
-
1
H
n
-
2
≤
⋯
≤
H
j
+
1
H
j
≤
⋯
≤
H
2
H
1
≤
H
1
,
where the equality holds only at the points k
1
=
k
2
=
⋯
=
k
n. From (79) we have
(91)
H
n
-
1
H
n
-
2
≥
H
n
H
n
-
1
=
∑
j
=
r
n
-
1
‍
a
j
H
j
H
n
-
1
≥
∑
j
=
r
n
-
1
‍
a
j
H
j
-
1
H
n
-
2
;
then from (89) we obtain that
(92)
H
n
-
1
≥
∑
j
=
r
n
-
1
‍
a
j
H
j
-
1
,
where the equality holds only at the points k
1
=
k
2
=
⋯
=
k
n. By using Corollary 5 we have that M is totally umbilical.



---

*Source: 974695-2014-03-18.xml*