241
Acta Math. Univ. Comenianae
Vol. LXX, 2(2001), pp. 241–249
COMPLETE SPACE-LIKE SUBMANIFOLDS IN DE SITTER
SPACE
X. LIU
Abstract. In this paper, we characterize the complete space-like submanifolds
4(n−1)c
with parallel mean curvature vector satisfying H 2 =
in the de Sitter space
n2
completely.
1. Introduction
Let Mpn+p (c) be an (n + p)-dimensional connected semi-Riemannian manifold of
constant 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 as a de Sitter
space of index p, denote it by Spn+p (c). The study of space-like hypersurfaces in
de Sitter space has been recently of substantial interest from both physics and
mathematical points of view. Akutagawa [1] and Ramanathan [10] investigated
space-like hypersurfaces in a de Sitter space and proved independently that a
complete space-like hypersurface in a de Sitter space with constant mean curvature
is totally umbilical if the mean curvature H satisfies H 2 ≤ c when n = 2 and
n2 H 2 < 4(n − 1)c when n ≥ 3. Later, Cheng [3] generalized this result to general
submanifolds in a de Sitter space.
when n > 2
On the other hand, the well-known examples with H 2 = 4(n−1)c
n2
2
1
n−1
are umbilical sphere S n ( (n−2)
(c2 ),
n2 c) and the hyperbolic cylinder H (c1 ) × S
n−2
c1 = (2−n)c and c2 = n−1 c. Hence it is natural to study if complete space-like hy-
persurfaces with H 2 = 4(n−1)c
(n > 2) are only the above examples. In [4], Cheng
n2
gave an affirmative answer if M n is compact and gave some characterizations when
M n is complete and noncompact. In this paper, we consider the case of space-like
submanifolds with parallel mean curvature vector satisfying H 2 = 4(n−1)c
in the
n2
de Sitter space and prove the following theorem
Theorem. Let M n be an n-dimensional (n ≥ 3) complete space-like submanifold in the de Sitter space Spn+p (c) with parallel mean curvature vector. If
Received February 19, 2001.
2000 Mathematics Subject Classification. Primary 53C40, 53C42, 53A10.
Key words and phrases. Space-like submanifold, hyperbolic cylinder, parallel mean curvature
vector.
This work is supported in part by the National Natural Science Foundation of China.
242
X. LIU
n
H 2 = 4(n−1)
is totally umbilical, or M n is the hyperbolic cylinder
n2 c, then M
n+1
1
n−1
n
H (c1 ) × S
(c
R 2 ) inmS1 (c), or M has unbounded volume and positive Ricci
curvature and M n S dv = ∞ for any m, where S is the norm square of the
second fundamental form of M n .
2. Preliminaries
Let Spn+p (c) be an (n + p)-dimensional de Sitter space of constant curvature c
whose index is p. Let M n be an n-dimensional Riemannian manifold immersed
in Spn+p (c). As the semi-Riemannian metric of Spn+p (c) induces the Riemannian
metric of M n , M n is called a space-like submanifold. We choose a local field of
semi-Riemannian orthonormal frames e1 , . . . , en+p in Spn+p (c) such that at each
point of M n , e1 , . . . , en span the tangent space of M n 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
that the semi-Riemannian metric of
P frame
Pfield so P
2
Spn+p (c) is given by ds̄2 = i ωi2 − α ωα2 = A A ωA
, where i = 1 and α = −1.
n+p
Then the structure equations of Sp (c) are given by
X
(1)
dωA =
B ωAB ∧ ωB , ωAB + ωBA = 0,
B
(2)
dωAB =
X
C ωAC ∧ ωCB −
C
(3)
1X
KABCD ωC ∧ ωD ,
2
C,D
KABCD = c A B (δAC δBD − δAD δBC ).
Restrict these form to M n , we have
(4)
n + 1 ≤ α ≤ n + p,
P
is written as ds2 = i ωi2 . From Cartan’s lemma
ωα = 0,
the Riemannian metric of M n
we can write
(5)
ωαi =
X
α
α
hα
ij ωj , hij = hji .
j
From these formulas, we obtain the structure equations of M n :
X
(6)
dωi =
ωij ∧ ωj , ωij + ωji = 0,
j
(7)
(8)
1X
Rijkl ωk ∧ ωl ,
2
k
k,l
X
α
α α
= c(δik δjl − δil δjk ) −
(hα
ik hjl − hil hjk ),
dωij =
Rijkl
X
ωik ∧ ωkj −
α
COMPLETE SPACE-LIKE SUBMANIFOLDS IN DE SITTER SPACE
243
where Rijkl are the components of the curvature tensor of M n and
X
X
(9)
h=
hα eα =
hα
ij ωi ⊗ ωj ⊗ eα
α
i,j,α
is the second fundamental form of M n .
For indefinite Riemannian manifolds in detail, refer to O’Neill [7].
Let S be the norm square of the second fundamental form of M n , ξ denote the
mean curvature vector field of M n and H the mean curvature of M n , that is
X
1 X X α 2
ξ=
hii eα , H = |ξ|, S =
(hα
ij ) .
n α
i
α,i,j
Moreover, the normal curvature tensor {Rαβkl }, the Ricci curvature tensor
{Rik } and the scalar curvature R are expressed as
X
β
β
α
Rαβkl =
(hα
km hml − hlm hmk ),
m
Rik = (n − 1) c δik −
(10)
XX
X
α
α
(
hα
hα
ll )hik +
ij hjk ,
α
α,j
l
R = n(n − 1)c + (S − n2 H 2 ).
(11)
α
Define the first and the second covariant derivatives of {hα
ij }, say {hijk } and
{hα
ijkl } by
X
X
X
X β
α
(12)
hα
hα
hα
hij ωβα ,
ijk ωk = dhij +
kj ωki +
ik ωkj +
k
(13)
X
l
α
hα
ijkl ωl = dhijk +
k
X
k
hα
mjk ωmi +
m
X
hα
imk ωmj +
m
β
X
hα
ijm ωmk +
m
X
hβijk ωβα .
β
Then, by exterior differentiation of (5), we obtain the Codazzi equation
α
hα
ijk = hikj .
(14)
It follows that the Ricci identities hold
X
X
X β
α
hα
(15)
hα
hα
hij Rβαkl .
ijkl − hijlk =
mj Rmikl +
im Rmjkl +
m
m
β
α
The Laplacian ∆hα
ij of the fundamental form hij is defined to be
(15) we have
X
X
X
(16)
∆hα
hα
hα
hα
ij =
im Rmkjk +
mk Rmijk +
kkij .
m,k
m,k
P
k
hα
ijkk , from
k
We need the following generalized maximum principle due to Omori [9] and
Yau [11]:
244
X. LIU
Lemma 2.1. Let M n be an n-dimensional complete Riemannian manifold
whose Ricci curvature is bounded from below and F : M n → R a smooth function bounded from below. Then there is a sequence of points {pk } in M n such
that
lim F (pk ) = inf(F ), lim |∇F (pk )| = 0, lim inf ∆F (pk ) ≥ 0.
k→∞
k→∞
k→∞
We also need the following algebraic lemma due to M. Okumura [8] (see also [2]).
P
2.2. Let µi , i = 1, . . . , n, be real numbers such that i µi = 0 and
P Lemma
2
2
i µi = β , where β = constant ≥ 0. Then
X
n−2
n−2
(17)
β3 ≤
µ3i ≤ p
β3,
−p
n(n − 1)
n(n
−
1)
i
and the equality holds in (17) if and only if at least (n − 1) of the µi are equal.
Now we assume that the mean curvature vector ξ is parallel and H 2 =
We can choose en+1 = ξ/H. Then
X
α
(18)
kkki
= 0, ωα,n+1 = 0, H α H n+1 = H n+1 H α ,
4(n−1)
n2 c.
k
(19)
tr H n+1 = nH, tr H α = 0, α 6= n + 1,
where H α denote the matrix (hα
ij ).
Putting
(20)
α
µij = hn+1
− Hδij , τij
= hα
ij , α 6= n + 1,
ij
we have
(21)
(22)
|µ|2 = tr(µ)2 =
X
µ2ij = tr(H n+1 )2 − nH 2 ,
X
|τ |2 =
(hβij )2 ,
β6=n+1
(23)
tr µ = 0, tr(τ β ) = 0, β 6= n + 1,
(24)
S = |µ|2 + |τ |2 + nH 2 .
A submanifold M n is said to be pseudo-umbilical if it is umbilical with respect to the direction of the mean curvature vector ξ, i.e., hn+1
= Hδij . From
ij
(21)-(24) we know that M n is pseudo-umbilical if and only if |µ|2 = 0, M n is
totally umbilical if and only if |µ|2 = 0 and |τ |2 = 0.
X
X
β
β
β
β
(25)
∆hn+1
= nchn+1
− nHcδij +
hn+1
hn+1
ij
ij
km hmk hij −
km hmj hik
X
X
β
β
n+1
+
hn+1
hn+1
mi hmk hkj − nH
mi hmj .
Thus
(26)
X
X
1
2
2
2
2
∆(|µ|2 ) =
(hn+1
(hn+1
ij ) − n cH
ijk ) + nc
2
X
− nHtr(H n+1 )3 +
tr(H n+1 H β )2 + [tr(H n+1 )2 ]2 .
β6=n+1
COMPLETE SPACE-LIKE SUBMANIFOLDS IN DE SITTER SPACE
245
On the other hand
tr(H n+1 )3 = tr µ3 + 3H[tr(H n+1 )2 − nH 2 ] + nH 3 .
(27)
By using (23), (27) and Lemma 2.2, we have from (26)
1
∆(|µ|2 ) ≥ (|µ|2 + nH 2 )2 − nH[tr(µ)3 + 3H|µ|2 + nH 3 ] + nc|µ|2
2
= |µ|2 (|µ|2 + nc − nH 2 ) − nHtr(µ)3
!
n(n
−
2)
|H||µ| + nc − nH 2
≥ |µ|2 |µ|2 − p
n(n − 1)
2
(n − 2) √
2
= |µ| |µ| − √
c ,
n
(28)
where we used H 2 = 4(n−1)
n2 c.
Now consider the positive smooth function f on M n defined by
1
f=p
1 + |µ|2
.
It is easy to check that
|∇f |2 =
(29)
1 |∇(|µ|2 )|2
4 (1 + |µ|2 )3
and that
f ∆f = −
(30)
1 ∆(|µ|2 )
+ 3|∇f |2 .
2 (1 + |µ|2 )2
From (28) and (30), we have
√ √
f ∆f ≤ −|µ|2 (|µ| − (n − 2) c/ n)2 /(1 + |µ|2 )2 + 3|∇f |2 .
(31)
From (10) and H 2 =
(32)
4(n−1)
n2 c,
we have
X
p
2
Ric(ei ) ≥ (n − 1)c − nHhn+1
+
(hn+1
(n − 1)c) ≥ 0,
ii
ik ) = (λi −
k
hn+1
ij
where
= λi δij . So the Ricci curvature of M n is non-negative, we may apply
Lemma 2.1 to the smooth function f . Then there is a sequence of points pk in M n
such that
lim f (pk ) = inf f, lim |∇f (pk )| = 0, lim inf ∆f (pk ) ≥ 0.
k→∞
k→∞
k→∞
From (31), we have inf(f ) 6= 0, so limk→∞ |µ|2 (pk ) = sup |µ|2 < ∞. Approaching the limit of both sides of inequality (31), we obtain sup |µ|2 = 0, or
2
sup |µ|2 = (n−2)
c.
n
2
If |µ| reaches its supremum on M n , from (28) we know that |µ|2 is subharmonic.
Thus |µ|2 would be constant because of the maximum principle. So we have the
following proposition
246
X. LIU
Proposition 2.1. Let M n be an n-dimensional (n ≥ 3) complete space-like
submanifold in the de Sitter space Spn+p (c) with parallel mean curvature vector. If
H2 =
4(n−1)
n2 c,
then either M n is pseudo-umbilical or sup |µ|2 =
(n−2)2
c.
n
2
supremum is attained if and only if |µ| ≡
(n−2)2
c,
n
and this
3. The Proof of Theorem
By use of (18), we have from (16) for α 6= n + 1
X
X
β
β
β
β
α
(33)
∆hα
hα
hα
ij = nchij +
km hmk hij − 2
km hmj hik
X
X
X
β
β
β
β
n+1
+
hα
hα
hα
mi hmk hkj +
jm hmk hki − nH
mi hmj .
Thus
(34)
X
X
1
β
β α
2
2
∆(|τ |2 ) =
(hα
hα
km hmk hij hij
ijk ) + nc|τ | +
2
α6=n+1
α6=n+1
X
X
β
β α
β
β α
α
−2
hkm hmj hik hij +
hα
mi hmk hkj hij
α6=n+1
X
+
α6=n+1
β
β α
hα
jm hmk hki hij
X
− nH
α6=n+1
α n+1
hα
mi hij hmj .
α6=n+1
By use of (18) and (19), we have from (34)
X
1
2
2
(35)
∆(|τ |2 ) =
(hα
ijk ) + nc|τ | + I + II,
2
α6=n+1
where
(36)
X
I=
α,β6=n+1
β
β α
hα
mi hmk hkj hij
+
α,β6=n+1
(37)
II =
X
n+1 n+1 α
hα
km hmk hij hij − 2
β
β α
hα
jm hmk hki hij ,
X
n+1 n+1 α
hα
km hmj hik hij
X
n+1 n+1 α
hα
jm hmk hki hij
α6=n+1
X
n+1 n+1 α
hα
mi hmk hkj hij
α6=n+1
− nH
X
α,β6=n+1
α6=n+1
+
β
β α
hα
km hmj hik hij
α,β6=n+1
X
+
X
[tr(H α H β )]2 − 2
X
+
α6=n+1
α n+1
hα
mi hij hmj
α6=n+1
=
X
n+1 n+1 α
hα
km hmk hii hij − nH
α6=n+1
P
β
hα
ij hij
X
α n+1
hα
mi hij hmj .
α6=n+1
We put Sαβ =
for α, β 6= n + 1, then (Sαβ ) is a (p − 1) × (p − 1)
symmetric matrix. It can be assumed to be diagonal for a suitable choice of
COMPLETE SPACE-LIKE SUBMANIFOLDS IN DE SITTER SPACE
247
P
en+2 , . . . , en+p . Set Sα = Sαα and we have |τ |2 = α6=n+1 Sα . In general, for a
matrix A = (aij ), we put N (A) = tr(At A). Now we have from (36),
X
X
(38)
I=
Sα +
N (H α H β − H β H α )
α6=n+1
≥
X
α6=n+1
α,β6=n+1

Sα2 ≥ 
X
2
α6=n+1
Sα  /(p − 1) = |τ |4 /(p − 1).
By Proposition 2.1, we need to divide the proof of Theorem into the following
three cases.
Case (i): M n is pseudo-umbilical, that is |µ|2 = 0 or hn+1
= Hδij on M n ,
ij
from (37) we get
II = −nH 2 |τ |2 .
(39)
Thus, in this case, we have
1
(n − 2)2 2
∆(|τ |2 ) ≥ (nc − nH 2 )|τ |2 + |τ |4 /(p − 1) =
|τ | c + |τ |4 /(p − 1).
2
n
p
Let f = 1/ 1 + |τ |2 , by use of the similar methods of proof of |µ|2 in section
2, we have |τ |2 = 0. Hence M n is totally umbilical.
2
Case (ii): sup |µ|2 = (n−2)
c and supremum of |µ|2 is attained, then |µ|2 ≡
n
(40)
(n−2)2
c.
n
From Lemma 2.2, we have
√
p
c
λ1 = (n − 1)c, λ2 = · · · = λn = √
.
n−1
(41)
For any fixed α 6= n + 1, let hα
ij = αi δij , noting α1 + · · · + αn = 0, by use of
(41), we have for any α =
6 n+1
!2
2
X
X
√
1
n+1 n+1 α
α
(42)
hkm hmk hii hij =
λm αm
=c
n−1− √
α12 .
n
−
1
m
(43)
−nH
X
α n+1
hα
mi hij hmj = −nH
X
2
λm αm
m
√
√ √
= −nH c[ n − 1α12 + (α22 + · · · + αn2 )/ n − 1]
≥ −2c(n − 1)α12 − 2c(α22 + · · · + αn2 )
= −2c(n − 1)(x + (1 − x))α12 − 2c(α22 + · · · + αn2 )
≥ −2c(n − 1)xα12 − 2c(n − 1)(1 − x)(n − 1)(α22 + · · · + αn2 )
− 2c(α22 + · · · + αn2 )
= −2c(n − 1)xα12 − 2c[1 + (n − 1)2 (1 − x)](α22 + · · · + αn2 ),
where x is a real number satisfying 0 ≤ x ≤ 1.
248
X. LIU
Choosing x =
(44)
X
2n2 −5n+4
2(n−1)2 ,
for fixed α 6= n + 1, from (42) and (43) we have
n+1 n+1 α
hα
km hmk hii hij − nH
X
α n+1
hα
mi hij hmj
(n − 2)2
− 2(n − 1)x]cα12 − 2c[1 + (n − 1)2 (1 − x)](α22 + · · · + αn2 )
n−1
X
2
= −nc(α12 + · · · + αn2 ) = −nc
(hα
ij ) .
≥[
i,j
From (37), (43) and (44) we have
X
(45)
II ≥ −nc
2
2
(hα
ij ) = −nc|τ | .
i,j,α6=n+1
Combining (35) and (38) with (45), we get
(46)
1
|τ |4
∆(|τ |2 ) ≥
.
2
p−1
p
Let f = 1/ 1 + |τ |2 , by use of the similar methods of proof of |µ|2 in Section 2,
we have |τ |2 = 0. Hence M n is a hyperbolic cylinder H 1 (c1 )×S n−1 (c2 ) in S1n+1 (c).
2
2
Case (iii): sup |µ|2 = (n−2)
c and |µ|2 < (n−2)
c. By (32), we know that M n
n
n
has non-negative Ricci curvature. If there is a point p in M n and a unit vector
v ∈ Tp M n such that Ric(v, v)(p) = 0, then taking e1 = v, we obtain λi = nH
2 .
Hence
n2 H 2
(n − 2)2
|µ|2 =
+ λ22 + · · · + λ2n − nH 2 <
c,
4
n
namely
λ22 + · · · + λ2n < c.
Since
(n − 1)c =
n2 H 2
= (λ2 + · · · + λn )2 ,
4
we get
(n − 1)c > (n − 1)(λ22 + · · · + λ2n ) ≥ (λ2 + · · · + λn )2 = (n − 1)c.
This is a contradiction. Hence the Ricci curvature is positive. RFrom the result due
to Yau [12], we know that M n has unbounded volume and M n S m dv = ∞ for
any m. This completes the proof of Theorem.
Acknowledgements. This work was carried out during the author’s visit to
Max-Planck-Institut für Mathematik in Bonn. The author would like to exprss
his thanks to Professor Yuri Manin for the invitation and the staff of the MPIM
for very warm hospitality.
COMPLETE SPACE-LIKE SUBMANIFOLDS IN DE SITTER SPACE
249
References
1. Akutagawa K., On space-like hypersurfaces with constant mean curvature in the de Sitter
space, Math. Z. 196 (1987), 13–19.
2. Alencar H. and do Carmo M. P., Hypersurfaces with constant mean curvature in spheres,
Proc. Amer. Math. Soc. 120 (1994), 1223–1229.
3. Cheng Q. M., Complete space-like submanifolds in a de Sitter space with parallel mean
curvature vector, Math. Z. 206 (1991), 333–339.
4.
, Hypersurfaces of a Lorentz space form, Arch. Math. 63 (1994), 271–281.
5. Montiel S., An integral inequality for compact spacelike hypersurfaces in de Siter space and
applications to the case of constant mean curvature, Indiana Univ. Math. J. 37 (1988),
909–917.
, A characterization of hyperbolic cylinders in the de Sitter space, Tôhoku Math. J.
6.
48 (1996), 23–31.
7. O’Neill B., Semi-Riemannian Geometry, New York, Academic Press, 1983.
8. Okumuru M., Hypersurfaces and a pinching problem on the second fundamental tensor,
Amer. J. Math. 96 (1974), 207–213.
9. Omori H., Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan 19 (1967),
205–214.
10. Ramanathan J., Complete space-like hypersurfaces of constant mean curvature in the de
Sitter space, Indiana Univ. Math. J. 36 (1987), 349–359.
11. Yau S. T., Harmonic functions on complete Riemannian manifolds, Comm. Pure and Appl.
Math. 28 (1975), 201–228.
, Some function-theoretic properties of complete Riemannian manifolds and their
12.
applications to geometry, Indiana Univ. Math. J. 25 (1976), 659–670.
X. Liu, Department of Applied Mathematics, Dalian University of Technology, Dalian 116024,
China, e-mail: xmliu@dlut.edu.cn