Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field in de Sitter Space
Space-like submanifolds, with dimension greater than three and with negative definite normal bundle in a general de Sitter space, of any index, are studied. For the compact space-like submanifolds whose mean curvature has no zero and the corresponding normalized vector field is parallel, under natur...
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irk-123456789-1066892016-10-03T03:02:21Z Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field in de Sitter Space Shu, Sh. Space-like submanifolds, with dimension greater than three and with negative definite normal bundle in a general de Sitter space, of any index, are studied. For the compact space-like submanifolds whose mean curvature has no zero and the corresponding normalized vector field is parallel, under natural boundedness assumptions on the lengths of the gradient of the length of the mean curvature and the covariant derivative of the second fundamental form, it is proved that they must be totally umbilical. As an application, two characterizations of totally umbilical space-like submanifolds in terms of the scalar curvature and the length of its second fundamental form are given. All the results extend the previous ones obtained by Liu for the case of space-like hypersurfaces in de Sitter space of index one. In addition, for the complete space-like submanifolds, whose normalized mean curvature vector field is parallel, two characterizations of totally umbilical space-like submanifolds and hyperbolic cylinders are obtained. 2011 Article Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field in de Sitter Space / Sh. Shu // Журнал математической физики, анализа, геометрии. — 2011. — Т. 7, № 4. — С. 352-369. — Бібліогр.: 22 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106689 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Space-like submanifolds, with dimension greater than three and with negative definite normal bundle in a general de Sitter space, of any index, are studied. For the compact space-like submanifolds whose mean curvature has no zero and the corresponding normalized vector field is parallel, under natural boundedness assumptions on the lengths of the gradient of the length of the mean curvature and the covariant derivative of the second fundamental form, it is proved that they must be totally umbilical. As an application, two characterizations of totally umbilical space-like submanifolds in terms of the scalar curvature and the length of its second fundamental form are given. All the results extend the previous ones obtained by Liu for the case of space-like hypersurfaces in de Sitter space of index one. In addition, for the complete space-like submanifolds, whose normalized mean curvature vector field is parallel, two characterizations of totally umbilical space-like submanifolds and hyperbolic cylinders are obtained. |
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Shu, Sh. |
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Shu, Sh. Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field in de Sitter Space Журнал математической физики, анализа, геометрии |
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Shu, Sh. |
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Shu, Sh. |
| title |
Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field in de Sitter Space |
| title_short |
Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field in de Sitter Space |
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Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field in de Sitter Space |
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Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field in de Sitter Space |
| title_full_unstemmed |
Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field in de Sitter Space |
| title_sort |
space-like submanifolds with parallel normalized mean curvature vector field in de sitter space |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/106689 |
| citation_txt |
Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field in de Sitter Space / Sh. Shu // Журнал математической физики, анализа, геометрии. — 2011. — Т. 7, № 4. — С. 352-369. — Бібліогр.: 22 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
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AT shush spacelikesubmanifoldswithparallelnormalizedmeancurvaturevectorfieldindesitterspace |
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2025-07-07T18:52:15Z |
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2025-07-07T18:52:15Z |
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1837015326478303232 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2011, vol. 7, No. 4, pp. 352–369
Space-like Submanifolds with Parallel Normalized Mean
Curvature Vector Field in de Sitter Space
Shichang Shu
Department of Mathematics, Xianyang Normal University
Xianyang, Shaanxi 712000, P. R. China
E-mail: shushichang@126.com
Received October 3, 2008
Space-like submanifolds, with dimension greater than three and with
negative definite normal bundle in a general de Sitter space, of any index,
are studied. For the compact space-like submanifolds whose mean curvature
has no zero and the corresponding normalized vector field is parallel, under
natural boundedness assumptions on the lengths of the gradient of the length
of the mean curvature and the covariant derivative of the second fundamental
form, it is proved that they must be totally umbilical. As an application,
two characterizations of totally umbilical space-like submanifolds in terms
of the scalar curvature and the length of its second fundamental form are
given. All the results extend the previous ones obtained by Liu for the case
of space-like hypersurfaces in de Sitter space of index one. In addition,
for the complete space-like submanifolds, whose normalized mean curvature
vector field is parallel, two characterizations of totally umbilical space-like
submanifolds and hyperbolic cylinders are obtained.
Key words: space-like submanifold, de Sitter space, normalized mean
curvature vector, totally umbilical submanifold.
Mathematics Subject Classification 2000: 53C40, 53C42.
1. Introduction
Let Mn+p
p (c) be an (n+p)-dimensional connected semi-Riemannian manifold
of constant sectional curvature c whose index is p. It is called an indefinite space
form of index p and simply a space form when p = 0. If c > 0, we call it a de Sitter
space of index p and denote by Sn+p
p (c). It was pointed out by Marsden and Tipler
[1] and Stumbles [2] that space-like hypersurfaces with constant mean curvature
Project supported by NSF of Shaanxi Province (SJ08A31) and NSF of Shaanxi Educational
Committee (11JK0479).
c© Shichang Shu, 2011
Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field
in arbitrary space-time got interesting in the relativity theory. Space-like hyper-
surfaces with constant mean curvature are convenient as initial hypersurfaces for
the Cauchy problem in arbitrary space-time and for studying the propagation of
gravitational radiation. Therefore, space-like hypersurfaces in a de Sitter space
with constant mean curvature have recently been studied by many differential
geometers from both physics and mathematical points of view. For example, one
can see [3–6]. Goddard [4] conjectured that the complete constant mean curva-
ture space-like hypersurfaces in a de Sitter space must be umbilical. Akutagawa
[3] and Ramanathan [6] proved independently that a complete space-like hyper-
surface in a de Sitter space with constant mean curvature is totally umbilical if
the mean curvature H satisfies H2 ≤ c when n = 2 and n2H2 < 4(n− 1)c when
n ≥ 3. The well-known examples with H2 = 4(n−1)/n2 are the umbilical sphere
Sn((n − 2)2/n2)) and the hyperbolic cylinder H1(c1) × Sn−1(c2), c1 = (2 − n)
and c2 = (n− 2)/(n− 1). Later, Cheng [7] generalized the result of [3] and [6] to
general submanifolds with higher codimension in a de Sitter space Sn+p
p (c).
On the other hand, there are some interesting results related to the study
of space-like hypersurfaces in a de Sitter space with constant scalar curvature,
see, for instance [8–10]. Recently, Camargo, Chaves and De Sousa Jr. [11] have
studied the complete space-like submanifolds with higher codimension in a de
Sitter space Sn+p
p (c). If the normalized mean curvature vector field is parallel,
the scalar curvature n(n−1)R is constant and R ≤ c, they obtain some interesting
results.
We should notice that the investigation on space-like hypersurfaces with the
scalar curvature n(n−1)R and the mean curvature H being linearly related is also
interesting, see, for instance, [8, 9, 12, 13]. Cheng [12] and Li [8] obtained some
characteristic theorems of such hypersurfaces in terms of the sectional curvature.
Recently, the author [13] proved a characteristic theorem of such hypersurfaces
in terms of the mean curvature H. The well-known complete space-like hyper-
surfaces with constant mean curvature are given by
Mn = {p ∈ Sn+1
1 | p2
k+1 + · · ·+ p2
n+1 = cosh2 r},
with r ∈ R1 and 1 ≤ k ≤ n, where R1 is the set of all real numbers. We can
prove that Mn is isometric to the Riemannian product Hk(sinh r)×Sn−k(cosh r)
of a k-dimensional hyperbolic space and a (n − k)-dimensional sphere of radii
sinh r and cosh r, respectively. Mn has k principal curvatures equal to coth r and
(n − k) principal curvatures equal to tanh r, so the mean curvature is given by
nH = k coth r + (n − k) tanh r. If k = 1, the Riemannian product H1(sinh r) ×
Sn−1(cosh r) is called a hyperbolic cylinder.
Let |∇h|2 =
∑
i,j,k,α(hα
ijk)
2 and |∇H|2 =
∑
i,α(Hα
,i )
2. From Proposition 3.1
and Proposition 3.2 in Section 3, we should notice that the condition |∇h|2 ≥
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 353
Shichang Shu
n2|∇H|2 is the natural generalization of one of the following three conditions: (i)
H = constant, (ii) the scalar curvature n(n − 1)R is constant and R ≤ c, (iii)
the scalar curvature n(n− 1)R is proportional to the mean curvature H, that is,
n(n− 1)R = kH.
For compact space-like hypersurfaces in a de Sitter space Sn+1
1 (1) with |∇h|2 ≥
n2|∇H|2, Liu [13] has recently proved the following results:
Theorem 1.1. Let Mn be an n-dimensional (n ≥ 3) compact space-like hy-
persurface in an (n+1)-dimensional de Sitter space Sn+1
1 (1). If |∇h|2 ≥ n2|∇H|2
and
|h|2 ≤ 2
√
n− 1,
then Mn is a totally umbilical hypersurface, where |h|2 is the squared norm of the
second fundamental form and H is the mean curvature of Mn.
Corollary 1.1. Let Mn be an n-dimensional (n ≥ 3) compact space-like
hypersurface with constant scalar curvature n(n− 1)R in an (n + 1)-dimensional
de Sitter space Sn+1
1 (1). If R ≤ 1 and
|h|2 ≤ 2
√
n− 1,
then Mn is a totally umbilical hypersurface.
Corollary 1.2. Let Mn be an n-dimensional (n ≥ 3) compact space-like
hypersurface in an (n+1)-dimensional de Sitter space Sn+1
1 (1). Suppose that the
scalar curvature n(n− 1)R is proportional to the mean curvature H of Mn, that
is, there exists a constant k such that n(n− 1)R = kH. If
|h|2 ≤ 2
√
n− 1,
then Mn is a totally umbilical hypersurface.
It is natural and interesting to study the n-dimensional compact space-like
submanifolds in a de Sitter space Sn+p
p (c) with |∇h|2 ≥ n2|∇H|2. We should
point out that the normalized mean curvature vector field is defined by ξ
H , where
ξ and H denote the mean curvature vector field and the mean curvature of Mn,
respectively. It is well known that submanifolds with nonzero parallel mean cur-
vature vector field also have parallel normalized mean curvature vector field. The
condition to have parallel normalized mean curvature vector field is much weaker
than the condition to have parallel mean curvature vector field. If the mean cur-
vature vector field is parallel, that is, ∇H = 0, we have H constant.
354 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field
In this paper, by using Cheng-Yau’s self-adjoint operator, we generalize Liu’s
results to general submanifolds in a de Sitter space Sn+p
p (c) with parallel normal-
ized mean curvature vector field. We shall prove the following:
Theorem 1.2. Let Mn be an n-dimensional (n ≥ 3) compact space-like
submanifold in an (n + p)-dimensional de Sitter space Sn+p
p (c). Suppose that the
normalized mean curvature vector field is parallel. If |∇h|2 ≥ n2|∇H|2 and
|h|2 ≤ nc/[(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
],
then Mn is a totally umbilical submanifold, where |h|2 is the squared norm of the
second fundamental form and H is the mean curvature of Mn.
Since we know that submanifolds with nonzero parallel mean curvature vector
field also have parallel normalized mean curvature vector field and ∇H = 0, we
can easily see that
Corollary 1.3. Let Mn be an n-dimensional (n ≥ 3) compact space-like
submanifold with nonzero parallel mean curvature vector field in an (n + p)-
dimensional de Sitter space Sn+p
p (c). If
|h|2 ≤ nc/[(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
],
then Mn is a totally umbilical submanifold.
We also have the following:
Corollary 1.4. Let Mn be an n-dimensional (n ≥ 3) compact space-like
submanifold with constant scalar curvature n(n− 1)R in an (n + p)-dimensional
de Sitter space Sn+p
p (c). Suppose that the normalized mean curvature vector field
is parallel. If R ≤ c and
|h|2 ≤ nc/[(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
],
then Mn is a totally umbilical submanifold.
Corollary 1.5. Let Mn be an n-dimensional (n ≥ 3) compact space-like
submanifold in an (n + p)-dimensional de Sitter space Sn+p
p (c). Suppose that
the normalized mean curvature vector field is parallel and the scalar curvature
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 355
Shichang Shu
n(n− 1)R is proportional to the mean curvature H of Mn, that is, there exists a
constant k such that n(n− 1)R = kH. If
|h|2 ≤ nc/[(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
],
then Mn is a totally umbilical submanifold.
R e m a r k 1.1. If p = 1 and c = 1, we have
|h|2 ≤ nc/[(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
] = 2
√
n− 1,
then Theorem 1.2, Corollary 1.3 and Corollary 1.4 reduce to Theorem 1.1, Corol-
lary 1.1 and Corollary 1.2, respectively. Therefore, we generalize the previous
results obtained by Liu [9] to general submanifolds with higher codimension.
R e m a r k 1.2. We should notice that L.J. Alias and A. Romero [14] proved
an integral formula for the compact space-like n-submanifolds in de Sitter spaces
Sn+p
q (c), 1 ≤ q ≤ p, by calculating the divergence of certain tangent vector fields
and using the divergence theorem. They obtained a Bernstein type result for the
complete maximal submanifolds in Sn+p
q (c), 1 ≤ q ≤ p. From [15], if p = q, we
know that the complete maximal space-like submanifolds in Sn+p
p (c) or Rn+p
p are
totally geodesic. Therefore, the class of all these submanifolds is very small. But
if q < p, we see that the class of complete maximal space-like submanifolds is very
large (see [16]). Thus, it is very interesting to study the n-dimensional space-like
submanifolds in Sn+p
q (c), 1 ≤ q < p. The Simons’ formulas of the n-dimensional
space-like submanifolds in Sn+p
q (c), 1 ≤ q < p, from those in Sn+p
p (c). Thus, the
results will be different.
2. Preliminary
Let Sn+p
p (c) be an (n + p)-dimensional de Sitter space with index p. Let Mn
be an n-dimensional connected space-like submanifold immersed in Sn+p
p (c). We
choose a local field of the semi-Riemannian orthonormal frames e1, . . . , en+p in
Sn+p
p (c) such that at each point of Mn, e1, . . . , en span the tangent space of Mn
and form an orthonormal frame there. We use the following convention on the
range of indices:
1 ≤ A,B, C, . . . ≤ n + p; 1 ≤ i, j, k, . . . ≤ n, n + 1 ≤ α, β, γ, . . . ≤ n + p.
Let ω1, . . . , ωn+p be its dual frame field so that the semi-Riemannian metric of
Sn+p
p (c) is given by ds2 =
∑
i
ω2
i −
∑
α
ω2
α =
∑
A
εAω2
A, where εi = 1 and εα = −1.
356 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field
Then the structure equations of Sn+p
p (c) are given by
dωA =
∑
B
εBωAB ∧ ωB, ωAB + ωBA = 0, (2.1)
dωAB =
∑
C
εCωAC ∧ ωCB − 1
2
∑
C,D
εCεDKABCDωC ∧ ωD, (2.2)
KABCD = cεAεB(δACδBD − δADδBC). (2.3)
If we restrict these form to Mn, then
ωα = 0, n + 1 ≤ α ≤ n + p. (2.4)
From Cartan’s lemma we have
ωαi =
∑
j
hα
ijωj , hα
ij = hα
ji. (2.5)
The connection forms of Mn are characterized by the structure equations
dωi =
n∑
j=1
ωij ∧ ωj , ωij + ωji = 0, (2.6)
dωij =
∑
k
ωik ∧ ωkj − 1
2
∑
k,l
Rijklωk ∧ ωl, (2.7)
Rijkl = c(δikδjl − δilδjk)−
∑
α
(hα
ikh
α
jl − hα
ilh
α
jk), (2.8)
where Rijkl are the components of the curvature tensor of Mn.
Denote by h the second fundamental form of Mn. Then
h =
∑
i,j,α
hα
ijωi ⊗ ωj ⊗ eα. (2.9)
Denote by ξ,H and |h|2 the mean curvature vector field, the mean curvature
and the squared norm of the second fundamental form of Mn, respectively. Then
they are defined by
ξ =
1
n
∑
α
(
∑
i
hα
ii)eα, H = |ξ| = 1
n
√∑
α
(
∑
i
hα
ii)2, |h|2 =
∑
i,j,α
(hα
ij)
2. (2.10)
Moreover, the normal curvature tensor Rαβkl, the Ricci curvature tensor Rik
and the scalar curvature n(n− 1)R are expressed as
Rαβkl =
∑
m
(hα
kmhβ
ml − hα
lmhβ
mk), (2.11)
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 357
Shichang Shu
Rik = (n− 1)cδik −
∑
α
(
∑
l
hα
ll)h
α
ik +
∑
α,j
hα
ijh
α
jk, (2.12)
n(n− 1)R = n(n− 1)c + |h|2 − n2H2, (2.13)
where R is the normalized scalar curvature.
Define the first and the second covariant derivatives of hα
ij , say hα
ijk and hα
ijkl,
by ∑
k
hα
ijkωk = dhα
ij +
∑
k
hα
ikωkj +
∑
k
hα
jkωki −
∑
β
hβ
ijωβα, (2.14)
∑
l
hα
ijklωl = dhα
ijk +
∑
m
hα
mjkωmi +
∑
m
hα
imkωmj +
∑
m
hα
ijmωmk −
∑
β
hβ
ijkωβα.
(2.15)
We obtain the Codazzi equation by straightforward computations
hα
ijk = hα
ikj . (2.16)
It follows that the Ricci identities hold
hα
ijkl − hα
ijlk =
∑
m
hα
imRmjkl +
∑
m
hα
jmRmikl +
∑
β
hβ
ijRαβkl. (2.17)
The Laplacian of hα
ij is defined by ∆hα
ij =
∑
k
hα
ijkk. From (2.17), for any
α, n + 1 ≤ α ≤ n + p, we obtain
∆hα
ij =
∑
k
hα
kkij +
∑
k,m
hα
kmRmijk +
∑
k,m
hα
imRmkjk +
∑
k,β
hβ
ikRαβjk. (2.18)
In the case when the mean curvature vector ξ has no zero, we know that ξ/H is
a normal vector field defined globally on Mn. We define |µ|2 and |τ |2 by
|µ|2 =
∑
i,j
(hn+1
ij −Hδij)2, |τ |2 =
∑
α>n+1
∑
i,j
(hα
ij)
2, (2.19)
respectively. Then |µ|2 and |τ |2 are functions defined on Mn globally, which do
not depend on the choice of the orthonormal frame {e1, . . . , en}. We have
|h|2 = nH2 + |µ|2 + |τ |2. (2.20)
Since the normalized mean curvature vector field is parallel, we choose en+1 =
ξ/H. Then
trHn+1 =
∑
i
hn+1
ii = nH, trHα =
∑
i
hα
ii = 0 (α ≥ n + 2). (2.21)
358 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field
From (2.8), (2.11), (2.18) and (2.21), by direct calculation we get (see [11] )
1
2
∆|h|2 =
∑
i,j,k,α
(hα
ijk)
2 +
∑
i,j
hn+1
ij (nH)ij + nc(|h|2 − nH2) (2.22)
− nH
∑
α
tr(H2
αHn+1) +
∑
α,β
[tr(HαHβ)]2
+
∑
α,β
N(HαHβ −HβHα),
where Hα denotes the matrix (hα
ij) for all α, N(A) = tr(AAt) for any matrix
A = (aij).
We need the following lemma
Lemma 2.1 ([17]). Let A,B be symmetric n× n matrices satisfying AB =
BA and trA = trB = 0. Then
|trA2B| ≤ n− 2√
n(n− 1)
(trA2)(trB2)1/2, (2.23)
and the equality holds if and only if (n − 1) of the eigenvalues xi of B and the
corresponding eigenvalues yi of A satisfy |xi| = (trB2)1/2/
√
n(n− 1), xixj ≥ 0,
yi = (trA2)1/2/
√
n(n− 1).
3. Proof of Theorem
For a C2-function f defined on Mn, we define its gradient and Hessian (fij)
by df =
∑
i
fiωi,
∑
j
fijωj = dfi +
∑
j
fjωji. Let T =
∑
i,j
Tijωi⊗ωj be a symmetric
tensor on Mn defined by Tij = nHδij − hn+1
ij . Following Cheng-Yau [18], we
introduce an operator ¤ associated to T acting on f by
¤f =
∑
i,j
Tijfij =
∑
i,j
(nHδij − hn+1
ij )fij . (3.1)
Since Mn is compact, the operator ¤ is self-adjoint (see [18]) if and only if
∫
M
(¤f)gdv =
∫
M
f(¤g)dv,
where f and g are any smooth functions on Mn.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 359
Shichang Shu
By a simple calculation and from (2.13), we obtain
¤(nH) =
∑
i,j
(nHδij − hn+1
ij )(nH)ij (3.2)
=
1
2
∆(n2H2)− n2|∇H|2 −
∑
i,j
hn+1
ij (nH)ij
=− 1
2
n(n− 1)∆R +
1
2
∆|h|2 − n2|∇H|2 −
∑
i,j
hn+1
ij (nH)ij .
Set φα
ij = hα
ij − 1
ntrHαδij and consider the symmetric tensor φ =
∑
i,j,α
φα
ijωiωjeα.
We can easily know that φ is traceless and
N(Φα) = N(Hα)− 1
n
(trHα)2, |φ|2 =
∑
α
N(Φα) = |h|2 − nH2, (3.3)
where Φα denotes the matrix (φα
ij).
Since the normalized mean curvature vector field is parallel, choosing en+1 =
ξ/H, from (2.21), we infer that
φn+1
ij = hn+1
ij −Hδij , φα
ij = hα
ij , (α ≥ n + 2),
N(Φn+1) = N(Hn+1)− nH2, N(Φα) = N(Hα), (α ≥ n + 2), (3.4)
tr(Hn+1)3 = tr(Φn+1)3 + 3HN(Φn+1) + nH3.
From (2.22), (3.3) and (3.4), we have
1
2
∆|h|2 ≥
∑
i,j,k,α
(hα
ijk)
2 +
∑
i,j
hn+1
ij (nH)ij + n(c−H2)|φ|2 (3.5)
− nH
∑
α
tr(Φ2
αΦn+1) +
∑
α,β
[tr(ΦαΦβ)]2.
Since we choose en+1 = ξ/H, we have ωαn+1 = 0 for all α. Consequently,
Rαn+1jk = 0, from (2.11), we have
∑
i
hα
ijh
n+1
ik =
∑
i
hα
ikh
n+1
ij , that is, HαHn+1 =
Hn+1Hα. Thus ΦαΦn+1 = Φn+1Φα. Since the matrices Φα and Φn+1 are trace-
less, by Lemma 2.1, we have
∑
α
tr(Φ2
αΦn+1) ≤ n− 2√
n(n− 1)
|µ||φ|2 ≤ n− 2√
n(n− 1)
|φ|3, (3.6)
where the following
|µ|2 ≤ |h|2 − nH2 = |φ|2 (3.7)
360 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field
is used. By the Cauchy–Schwarz inequality, we have
∑
α,β
[tr(ΦαΦβ)]2 ≥
∑
α
[N(Φα)]2 ≥ 1
p
|φ|4. (3.8)
From (3.5), (3.6) and (3.8), we have
1
2
∆|h|2 ≥
∑
i,j,k,α
(hα
ijk)
2 +
∑
i,j
hn+1
ij (nH)ij (3.9)
+ |φ|2{nc− nH2 − n(n− 2)√
n(n− 1)
H|φ|+ 1
p
|φ|2}.
From (3.2) and (3.9), we have
¤(nH) ≥− 1
2
n(n− 1)∆R + (|∇h|2 − n2|∇H|2) (3.10)
+ |φ|2{nc− nH2 − n(n− 2)√
n(n− 1)
H|φ|+ 1
p
|φ|2}.
P r o o f of Theorem 1.2. Since Mn is compact and the operator ¤ is
self-adjoint, by |∇h|2 ≥ n2|∇H|2 and Stokes formula, we have
0 ≥
∫
Mn
|φ|2{nc− nH2 − n(n− 2)√
n(n− 1)
H|φ|+ 1
p
|φ|2}dv (3.11)
=
∫
Mn
|φ|2PH(|φ|)dv,
where PH(|φ|) = n(c−H2)− n(n−2)√
n(n−1)
H|φ|+ 1
p |φ|2.
Considering the quadratic form Q(u, t) = 1
pu2 − n−2√
n−1
ut − t2 and by the
orthogonal transformation
ũ =
1√
2n
{(1 +
√
n− 1)u + (1−√n− 1)t},
t̃ =
1√
2n
{(√n− 1− 1)u + (
√
n− 1 + 1)t},
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 361
Shichang Shu
we have
Q(u, t) =
1
2n
{[1
p
(n + 2
√
n− 1) +
(n− 2)2√
n− 1
− (n− 2
√
n− 1)]ũ2
− 2(1− 1
p
)(n− 2)ũt̃ + [
1
p
(n− 2
√
n− 1)− (n− 2)2√
n− 1
− (n + 2
√
n− 1)]t̃2}
=− 1
2n
[
1
p
(2
√
n− 1− n) +
(n− 2)2√
n− 1
+ (n + 2
√
n− 1)](ũ2 + t̃2)
+
1
2n
[(
1
p
+ 1)4
√
n− 1 +
2(n− 2)2√
n− 1
]ũ2 − 1
2n
(1− 1
p
)(n− 2)2ũt̃
≥− 1
2n
[
1
p
(2
√
n− 1− n) +
(n− 2)2√
n− 1
+ (n + 2
√
n− 1)
+ (1− 1
p
)(n− 2)](ũ2 + t̃2) +
1
2n
[(
1
p
+ 1)4
√
n− 1 +
2(n− 2)2√
n− 1
]ũ2
=− [(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
](ũ2 + t̃2)
+
1
n
[(1 +
1
p
)2
√
n− 1 +
(n− 2)2√
n− 1
]ũ2
≥− [(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
](ũ2 + t̃2),
where ũ2 + t̃2 = u2 + t2.
Take u = |φ|, t =
√
nH, then
PH(|φ|) = nc + Q(u, t) ≥ nc− [(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
]|h|2.
From (3.11) and the assumption of Theorem 1.2, we have
0 ≥
∫
Mn
|φ|2{nc− [(1+
1
p
)
√
n− 1
n
+(1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
]|h|2}dv ≥ 0. (3.12)
Therefore, we see that
|φ|2{nc− [(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
]|h|2} = 0.
This implies that either |φ|2 = 0 or Mn is totally umbilical, or
nc− [(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
]|h|2 = 0.
362 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field
In the latter case, we infer that the equalities hold in (3.12), (3.11), (3.7) and
(2.23) of Lemma 2.1. If the equality holds in (3.7), we have |µ|2 = |h|2 − nH2,
this implies that |τ | = 0. Since en+1 is parallel on the normal bundle T⊥(Mn)
of Mn, by using the method of B.Y. Chen [19] or Yau [20], we know that Mn
lies in a totally geodesic submanifold Sn+1
1 (c) of Sn+p
p (c). If the equality holds in
Lemma 2.1, then (n− 1) of the numbers λi−H are equal to N(Φn+1)√
n(n−1)
= |µ|√
n(n−1)
,
or equal to the negative of this last expression, where λiδij = hn+1
ij . It follows
that Mn has at most two distinct constant principle curvatures. We conclude
that Mn is totally umbilical from the compactness of Mn. This completes the
proof of Theorem 1.2.
From [21], we have the following:
Proposition 3.1. Let Mn be an n-dimensional space-like submanifold in an
(n + p)-dimensional de Sitter space Sn+p
p (c). If the scalar curvature n(n− 1)R is
constant and R ≤ c, then we have
|∇h|2 ≥ n2|∇H|2.
We may also prove the following:
Proposition 3.2. Let Mn be an n-dimensional space-like submanifold in an
(n + p)-dimensional de Sitter space Sn+p
p (c). If the scalar curvature n(n− 1)R is
proportional to the mean curvature H of Mn, that is, there exists a constant k
such that n(n− 1)R = kH, then we have
|∇h|2 ≥ n2|∇H|2.
P r o o f. For a fixed α, we choose an orthonormal frame field {ei} at each
point on Mn so that hα
ij = λα
i δij . Then we have |h|2 =
∑
i,j,α
(hα
ij)
2 6= 0. In fact, if
|h|2 =
∑
i,α
(λα
i )2 = 0 at a point of Mn, then λα
i = 0 for all i and α at this point.
This implies H = 0 and R = 0 at this point. From (2.13), we have n(n−1)c = 0.
This is impossible. From (2.13) and n(n− 1)R = kH, we have
k∇iH = −2n2H∇iH + 2
∑
j,k,α
hα
kjh
α
kji,
(
k
2
+ n2H)2|∇H|2 =
∑
i
(
∑
j,k,α
hα
kjh
α
kji)
2 ≤
∑
i,j,α
(hα
ij)
2
∑
i,j,k,α
(hα
ijk)
2 = |h|2|∇h|2.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 363
Shichang Shu
Thus, we have
|∇h|2 − n2|∇H|2 ≥[(
k
2
+ n2H)2 − n2|h|2]|∇H|2 1
|h|2
=[
(k)2
4
+ n3(n− 1)c]|∇H|2 1
|h|2 ≥ 0.
The proof of Proposition 3.2 is completed.
From Theorem 1.2, Proposition 3.1 and Proposition 3.2, we can easily see
that Corollary 1.4 and Corollary 1.5 are true.
4. Some Related Results for Complete Cases
In this section, we study the complete space-like submanifolds in a de Sitter
space Sn+p
p (c) with parallel normalized mean curvature vector field. We obtain
the following:
Theorem 4.1. Let Mn be an n-dimensional (n ≥ 3) complete space-like
submanifold with constant scalar curvature n(n− 1)R in an (n + p)-dimensional
de Sitter space Sn+p
p (c). Suppose that the normalized mean curvature vector field
is parallel and the mean curvature H obtains its supremum on Mn. If R < c and
|h|2 ≤ nc/[(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
],
then Mn is totally umbilical, or Mn is isometric to a hyperbolic cylinder H1(sinh r)×
Sn−1(cosh r).
Theorem 4.2. Let Mn be an n-dimensional (n ≥ 3) complete space-like
submanifold in an (n + p)-dimensional de Sitter space Sn+p
p (c). Suppose that
the normalized mean curvature vector field is parallel and the mean curvature H
obtains its supremum on Mn. If there exists a constant k such that n(n− 1)R =
kH and
|h|2 ≤ nc/[(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
],
then Mn is totally umbilical, or Mn is isometric to a hyperbolic cylinder H1(sinh r)
×Sn−1(cosh r).
We prove the following Lemma:
364 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field
Lemma 4.1. Let Mn be an n-dimensional space-like submanifold in a de
Sitter space Sn+p
p (c). Then the following properties hold:
(i) if R < c, then the operator ¤ defined by (3.1) is elliptic;
(ii) if n(n − 1)R = kH and H > 0, then the operator L = ¤ + (k/2n)∆ is
elliptic.
P r o o f. (i) Choosing a local orthonormal frame field {e1, . . . , en} such that
hn+1
ij = λiδij , we get ¤f =
∑
i
(nH − λi)fii. Since R < c, from (2.13), we have
|h|2 < n2H2. If there is one i such that nH − λi ≤ 0, then n2H2 ≤ λ2
i ≤ |h|2.
This is a contradiction. Thus, we have nH − λi > 0 for any i and the operator
¤ is elliptic.
(ii) For a fixed α, we choose a local orthonormal frame field {e1, . . . , en} at
each point on Mn so that hα
ij = λα
i δij . From H > 0, nH =
∑
i
hn+1
ii and
∑
i
hα
ii = 0
for n + 2 ≤ α ≤ n + p on Mn, we have for any i:
(nH − λn+1
i + k/2n) =
∑
j
λn+1
j − λn+1
i (4.1)
+ (1/2)[
∑
j,α
(λα
j )2 − n2H2 + n(n− 1)c]/(nH)
≥
∑
j
λn+1
j − λn+1
i
+ (1/2)[
∑
j
(λn+1
j )2 − (
∑
j
λn+1
j )2 + n(n− 1)c]/(nH)
=[(
∑
j
λn+1
j )2 − λn+1
i (
∑
j
λn+1
j )
− (1/2)
∑
l 6=j
λn+1
l λn+1
j + (1/2)n(n− 1)c](nH)−1
=[
∑
j
(λn+1
j )2 + (1/2)
∑
l 6=j
λn+1
l λn+1
j
− λn+1
i (
∑
j
λn+1
j ) + (1/2)n(n− 1)c](nH)−1
=[
∑
j 6=i
(λn+1
j )2 + (1/2)
∑
l6=j
l,j 6=i
λn+1
l λn+1
j + (1/2)n(n− 1)c](nH)−1
=(1/2)[
∑
j 6=i
(λn+1
j )2 + (
∑
j 6=i
λn+1
j )2 + n(n− 1)c](nH)−1 > 0.
Thus, L is an elliptic operator. The proof of Lemma 4.1 is completed.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 365
Shichang Shu
From (3.10) and the proof of Theorem 1.2, we have
¤(nH) ≥− 1
2
n(n− 1)∆R + (|∇h|2 − n2|∇H|2) (4.2)
+ |φ|2{nc− [(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
]|h|2}.
P r o o f of Theorem 4.1. Since the scalar curvature n(n − 1)R is constant
and R < c, from Proposition 3.1, (4.2) and the assumption of Theorem 4.1, we
have
¤(nH) ≥ |φ|2{nc− [(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
]|h|2} ≥ 0.
(4.3)
Since H obtains its supremum on Mn and ¤ is elliptic, we see that H is constant.
Thus, from (4.3), we get
|φ|2{nc− [(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
]|h|2} = 0.
It follows that |φ|2 = 0, and Mn is totally umbilical, or
nc− [(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
]|h|2 = 0.
In the latter case, we know that the equalities hold in (4.3), (4.2), (3.7) and
(2.23) of Lemma 2.1. By the same method as in the proof of Theorem 1.2, we see
that Mn lies in a totally geodesic submanifold Sn+1
1 (c) of Sn+p
p (c) and has two
distinct constant principle curvatures. Therefore, we know that Mn is isometric
to a hyperbolic cylinder H1(sinh r)× Sn−1(cosh r) from the congruence theorem
in [22]. This completes the proof of Theorem 4.1.
P r o o f of Theorem 4.2. Applying the operator L = ¤ + (k/2n)∆ to nH
and by Proposition 3.2, (4.2) and the assumption of Theorem 4.2, we have
L(nH) =¤(nH) +
k
2n
∆(nH) = ¤(nH) +
1
2
n(n− 1)∆R (4.4)
≥|φ|2{nc− [(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
]|h|2} ≥ 0.
Since the normalized mean curvature vector field is parallel and H 6= 0, from
(2.10) it follows that H > 0. From Lemma 4.1, we know that L is elliptic as H
366 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field
obtains its supremum on Mn, we can see that H is constant. Thus, from (4.4),
we get
|φ|2{nc− [(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
]|h|2} = 0.
It follows that |φ|2 = 0, and Mn is totally umbilical, or
nc− [(1 +
1
p
)
√
n− 1
n
+ (1− 1
p
)
n− 1
n
+
(n− 2)2
2n
√
n− 1
]|h|2 = 0.
By the same method as in the proof of Theorem 4.1, we see that Theorem 4.2 is
true.
If c = 1 and p = 1, we can easily see that there holds the following:
Corollary 4.1. Let Mn be an n-dimensional (n ≥ 3) complete space-like
hypersurface with constant scalar curvature n(n− 1)R in an (n + 1)-dimensional
de Sitter space Sn+1
1 (1). Suppose that the mean curvature H obtains its supremum
on Mn. If R ≤ 1 and
|h|2 ≤ 2
√
n− 1,
then Mn is totally umbilical, or Mn is isometric to a hyperbolic cylinder H1(sinh r)×
Sn−1(cosh r).
Corollary 4.2. Let Mn be an n-dimensional (n ≥ 3) compact space-like
hypersurface in an (n+1)-dimensional de Sitter space Sn+1
1 (1). Suppose that the
mean curvature H obtains its supremum on Mn. If there exists a constant k such
that n(n− 1)R = kH and
|h|2 ≤ 2
√
n− 1,
then Mn is totally umbilical, or Mn is isometric to a hyperbolic cylinder H1(sinh r)×
Sn−1(cosh r).
Acknowledgment. The author would like to thank the referee for his/her
careful reading of the original manuscript and for making many valuable sugges-
tions and comments that have significantly improved the paper.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 367
Shichang Shu
References
[1] J. Marsden and F. Tipler, Maximal Hypersurfaces and Foliations of Constant Mean
Curvature in General Relativity. — Phys. Rep. 66 (1980), 109–139.
[2] S. Stumbles, Hypersurfaces of Constant Mean Extrinsic Curvature. — Ann. Phys.
133 (1980), 28–56.
[3] K. Akutagawa, On Space-Like Hypersurfaces with Constant Mean Curvature in a
de Sitter Space. — Math. Z. 196 (1987), 13–19.
[4] A.J. Goddard, Some Remarks on the Existence of Space-Like Hypersurfaces of
Constant Mean Curvature. — Math. Proc. Combrige Phil. Soc. 82 (1977), 489–495.
[5] S. Montiel, An Integral Inequality for Compact Space-Like Hypersurfaces in De
Sitter Space and Applications to the Case of Constant Mean Curvature. — Indiana
Univ. Math. J. 37 (1988), 909–917.
[6] J. Ramanathan, Complete Space-Like Hypersurfaces of Constant Mean Curvature
in the de Sitter Space. — Indiana Univ. Math. J. 36 (1987), 349–359.
[7] Q.M. Cheng, Complete Space-Like Submanifolds in a de Sitter Space with Parallel
Mean Curvature Vector. — Math. Z. 206 (1991), 333–339.
[8] H. Li, Global Rigidity Theorems of Hypersurfaces. — Ark. Mat. 35 (1997), 327–351.
[9] X.M. Liu, Space-Like Hypersurfaces in the de Sitter Spaces. — J. Math. Phys. 42
(2001), 3965–3972.
[10] Y. Zheng, Space-Like Hypersurfaces with Constant Scalar Curvature in the de Sitter
Spaces. — Diff. Geom. Appl. 6 (1996), 51–54.
[11] F.E.C Camargo, R.S.B. Chaves, and L.A.M. De Sousa Jr., New Characterizations
of Complete Space-Like Submanifolds in Semi-Riemannian Space Forms. — Kodai
Math. J. 32 (2009), 209–230.
[12] Q.M. Cheng, Complete Space-Like Hypersurfaces of a de Sitter Space with r = kH.
— Mem. Fac. Sci. Kyushu Univ. 44 (1990), 67–77.
[13] S.C. Shu, Complete Space-Like Hypersurfaces in a de Sitter Space. — Bull. Austral.
Math. Soc. 73 (2006), 9–16.
[14] L.J. Alias and A. Romero, Integral Formulas for Compact Space-Like n-
Submanifolds in de Sitter Spaces. Applications to the Parallel Mean Curvature
Vector Case. — Manuscripta Math. 87 (1995), 405–416.
[15] T. Ishihara, Maximal Space-Like Submanifolds of a Pseudo-Riemannian Space of
Constant Curvature. — Michigan Math. J. 35 (1988), 345–352.
[16] Q.M. Cheng and S. Ishikawa, Complete Maximal Space-Like Submanifolds. —
Kodai Math. J. 20 (1997), 208–217.
[17] W. Santos, Submanifolds with Parallel Mean Curvature Vector in Sphere. — Tôhoku
Math. J. 46 (1994), 403–415.
368 Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4
Space-like Submanifolds with Parallel Normalized Mean Curvature Vector Field
[18] S.Y. Cheng and S.T. Yau, Hypersurfaces with Constant Scalar Curvatrue. — Math.
Ann. 225 (1977), 195–204.
[19] B.Y. Chen, On the Surface with Parallel Mean Curvature Vector. — Indiana Univ.
Math. J. 22 (1973), 655–666.
[20] S.T. Yau, Submanifolds with Constant Mean Curvature. — Amer. J. Math. 96
(1974), 346-366.
[21] S.C. Shu and S.Y, Liu, Complete Space-Like Submanifolds with Constant Scalar
Curvature in a de Sitter Space. — Balkan J. Geom. Appl. 9 (2004), 82–91.
[22] N. Abe, N. Koike, and S. Yamaguch, Congruence Theorems for Proper Semi-
Riemannin Hypersurfaces in a Real Space Form. — Yokohama Math. J. 35 (1987),
123–136.
Journal of Mathematical Physics, Analysis, Geometry, 2011, vol. 7, No. 4 369
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