ARCHIVUM MATHEMATICUM (BRNO)
Tomus 46 (2010), 79–86
SPACELIKE SUBMANIFOLDS
IN INDEFINITE SPACE FORM Mpn+p (c)
Yingbo Han
Abstract. In this paper, we get an intrinsic inequality for spacelike submanifolds in indefinite space form Mpn+p (c), (c > 0). We also get some rigidity
theorems for such spacelike submanifolds.
1. Introduction
Mpn+p (c)
Let
be n + p-dimensional connected semi-Riemannian manifold of
constant curvature c whose index is p. It is called indefinite space form of index
p. Let M be an n-dimensional Riemannian manifold immersed in Mpn+p (c). The
semi-Riemannian metric of Mpn+p (c) induces the Riemannian metric of M , M is
called a spacelike submanifold. Spacelike submanifolds in indefinite space form
Mpn+p (c) have been of increasing interesting in the recent years. There are many
results about these submanifolds, for instance, Dong [3], Wu [6, 7], Liu[4]. In [5],
the authors got an intrinsic inequality for spacelike hypersurfaces in de Sitter space
form M1n+1 whose index is 1. In this note, we generalize the intrinsic inequality
for spacelike hypersurface of de Sitter space to spacelike submanifolds of indefinite
space form Mpn+p (c) with index p ≥ 1. From this inequality, we also get some
rigidity theorems for such spacelike submanifolds.
2. Preliminaries
We choose a local field of semi-Riemannian orthonormal frames {e1 , . . . , en , en+1 ,
. . . , en+p } in Mpn+p (c) such that, restricted to M n , e1 , . . . , en are tangent to M n .
Let ω1 , . . . , ωn be its dual P
frame field such that the semi-Riemannian metric of
n+p
n+p
2
Mp (c) is given by ds = A=1 A (ωA )2 , where i = 1, i = 1, . . . , n and α = −1,
2000 Mathematics Subject Classification: primary 53C42; secondary 53B30.
Key words and phrases: spacelike submanifolds, indefinite space form.
Supported by the National Natural Science Foundation of China No. 10971029.
Received August 11, 2009, revised October 2009. Editor O. Kowalski.
80
YINGBO HAN
α = n + 1, . . . , n + p. Then the structure equations of Mpn+p (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
CD
KABCD = cA B (δAC δBD − δAD δBC ) .
We restrict these forms to M n , then
α = n + 1, . . . , n + p ,
P
and the Riemannian metric of M n is written as ds2 = i ωi2 . Since
X
(5)
0 = dωα = −
ωα,i ∧ ωi ,
(4)
ωα = 0 ,
i
by Cartan’s lemma we may write
X
(6)
ωα,i =
hα
ij ωj ,
α
hα
ij = hji .
j
From these formulas, we obtain the structure equations of M n :
X
(7)
dωi = −
ωij ∧ ωj , ωij + ωji = 0 ,
j
(8)
dωij = −
X
ωik ∧ ωkj +
k
(9)
1X
Rijkl ωk ∧ ωl ,
2
Rijkl = c(δik δjl − δil δjk ) −
k,l
α
(hα
ik hjl
α
− hα
il hjk ) ,
where Rijkl are the components of curvature tensor of M n . We call
X
(10)
h=
hα
ij ωi ⊗ ωj ⊗ eα
i,j,α
P
the second fundamental form of M n . The mean curvature vector is H = i,α hα
ii eα =
P
P
P
α
2
α 2
2
α
α
H e , where H =
i hii . We denote |h| =
i,j,α (hij ) , and |H| =
Pα α α2
n
α (H ) . We call that M is maximal if its mean curvature field vanishes, i.e.
H = 0.
α
Let hα
ij,k and hij,kl denote the covariant derivative and the second covariant
α
α
derivative of hij . Then we have hα
ij,k = hik,j and
X β
X
X
α
hα
hij Rαβkl ,
hα
hα
mj Rmikl +
ij,kl − hij,lk =
im Rmjkl +
m
m
m
where Rαβkl are the components of the normal curvature tensor of M n , that is
X
β
β α
Rαβkl =
(hα
ik hil − hik hil ) .
i
n
If Rαβkl = 0 at point x of M we say that the normal bundle connection of M n
is flat at x and it is well known [1] that Rαβkl = 0 at point x if and only if the
matrix (hα
ij ) are simultaneously diagonalizable at x.
SPACELIKE SUBMANIFOLDS IN INDEFINITE SPACE FORM Mpn+p (c)
81
3. Main results for space-like submanifolds
Lemma 3.1 (Cauchy-Swartz inequality). Let a1 , . . . , an ; b1 , . . . , bn be real numbers,
then
X
2 X X ai bi ≤
a2i
b2i
i
i
i
and the equality holds if and only if there exists a constant λ such that ai = λbi or
bi = λai , i = 1, . . . , n.
Theorem 3.2. If M n is a space-like submanifold of indefinite space form Mpn+p (c)
(c > 0), S and ρ are Ricci curvature tensor and the scalar curvature of M n ,
respectively, then
|S|2 ≥ 2cρ(n − 1) − c2 n(n − 1)2 .
(11)
Moreover, |S|2 = 2cρ(n − 1) − c2 n(n − 1)2 if and only if M n is a spacelike Einstein
submanifolds with S = c(n − 1)g, where g is the Riemannian metric of M n .
Proof. From the Gauss equation we get
o
Xn
X
X
α
α α
c(δkk δij − δil δjk ) −
(hα
Sij =
Rkikj =
kk hij − hik hjk )
k
(12)
α
k
= c(n − 1)δij −
X
H α hα
ij
+
α
X
α
hα
ik hjk
k,α
So
|S|2 =
X
2
Sij
=
ij
o2
Xn
X
X
α
c(n − 1)δij −
H α hα
hα
ij +
ik hjk
α
ij
k,α
X
2 X
2
Xn
α
=
c2 (n − 1)2 δij +
H α hα
+
hα
ij
ik hjk
α
ij
− 2c(n − 1)δij
X
k,α
H α hα
ij + 2c(n − 1)δij
X
α
α
hα
ik hjk
k,α
X
X
o
α α
−2
H α hα
h
h
ij
ik jk
α
k,α
= c2 n(n − 1)2 +
XX
ij
H α hα
ij
2
+
α
XX
ij
k,α
X
α
− 2c(n − 1)|H|2 + 2c(n − 1)
hα
ik hik
i,k,α
−2
XX
i,j
α
H α hα
ij
X
k,α
α
hα
ik hjk
α
hα
ik hjk
2
82
YINGBO HAN
and
ρ=
X
Sii =
o
Xn
X
X
α
c(n − 1) −
H α hα
hα
ii +
ik hik
i
α
i
k,α
X
2
= cn(n − 1) − |H| +
(hα
ij )
2
ijα
2
= cn(n − 1) − |H| + |h|2 ,
(13)
So
|S|2 = c2 n(n − 1)2 +
XX
H α hα
ij
2
+
XX
α
ij
ij
2
α
hα
ik hjk
k,α
2
− 2c(n − 1)|H| + 2c(n − 1) ρ + |H| − cn(n − 1)
X
XX
α α
−2
H α hα
h
h
ij
ik jk
i,j
α
XX
H α hα
ij
X
α
H α hα
ij
2
+
α
ij
XX
i,j
k,α
= 2cρ(n − 1) − c2 n(n − 1)2 +
−2
2
α
hα
ik hjk
XX
ij
α
hα
ik hjk
2
k,α
k,α
≥ 2cρ(n − 1) − c2 n(n − 1)2 +
XX
H α hα
ij
2
+
α
ij
XX
ij
α
hα
ik hjk
2
k,α
XX
2 1/2 X X
2 1/2
α
−2
H α hα
hα
ij
ik hjk
ij
α
ij
= 2cρ(n − 1) − c2 n(n − 1)2 +
k,α
n X X
ij
(14)
−
XX
ij
α
hα
ik hjk
2 1/2 o2
H α hα
ij
2 1/2
α
≥ 2cρ(n − 1) − c2 n(n − 1)2 .
k,α
The first inequality has used Lemma 3.1.
So we have
|S|2 ≥ 2cρ(n − 1) − c2 n(n − 1)2 .
Now we will prove the second part of this theorem.
If M n is a spacelike Einstein submanifold with S = c(n − 1)g, then we have the
following equations:
|S|2 = c2 n(n − 1)2 ,
and ρ = cn(n − 1) ,
i.e.
|S|2 = 2cρ(n − 1) − c2 n(n − 1)2 .
SPACELIKE SUBMANIFOLDS IN INDEFINITE SPACE FORM Mpn+p (c)
83
Conversely, if the Eq. (14) becomes an equality, then all the inequality of Eq. (14)
will become equality. From the Lemma 3.1, there exist a constant λ such that
X
X
α
H α hα
hα
ij = λ
ik hjk
α
k,α
or
λ
(15)
X
H α hα
ij =
α
X
α
hα
ik hjk
i, j ∈ {1, . . . , n}
for all
k,α
and
XX
(16)
H α hα
ij
2
=
XX
α
ij
ij
α
hα
ik hjk
2
.
k,α
(I) If λ = 0, we know that
X
X
α
(17)
H α hα
hα
ij = 0 or
ik hjk = 0 for all
α
i, j ∈ {1, . . . , n} .
k,α
then
H = 0 or
(18)
X
2
[hα
ik ] = 0
i,k,α
If H = 0, then M
equations:
n
is maximal. From the Eq. (14), we have the following
|S|2 = 2cρ(n − 1) − c2 n(n − 1)2 +
XX
H α hα
ij
2
+
α
ij
XX
ij
α
hα
ik hjk
2
k,α
XX
2 1/2 X X
2 1/2
α α
−2
H α hα
h
h
ik jk
ij
ij
(19)
α
ij
= 2cρ(n − 1) − c2 n(n − 1)2 +
k,α
XX
ij
α
hα
ik hjk
2
.
k,α
We have that
XX
(20)
ij
α
hα
ik hjk
2
= 0,
k,α
for i, j ∈ {1, . . . , n}. From this equation, we get
X
α
(21)
hα
i = 1, . . . , n .
ik hik = 0 for
k,α
n
So hα
is totally
ij = 0, for i, j ∈ {1, . . . , n} and α ∈ {n + 1, . . . , n + p}, i.e. M
geodesic.
P
2
α
If i,k,α [hα
ik ] = 0 ,so hij = 0, for i, j ∈ {1, . . . , n} and α ∈ {n + 1, . . . , n + p},
n
i.e. M is totally geodesic.
From the Eq. (12), we know that
(22)
Sij = c(n − 1)δij .
84
YINGBO HAN
P
P
α α
(II) If λ 6= 0, from the equation α H α hα
ij = λ
k,α hik hjk , and equation (16),
we have the following equation:
hXX
2 i
α
(23)
(λ2 − 1)
hα
= 0,
ik hjk
ij
P
α α 2
k,α hik hjk
P
α α 2
k,α hik hjk
P
k,α
2
= 0 or λ = 1.
P
α α 2
If ij
= 0, then
= 0 for all i, j. So hα
ij = 0, for
k,α hik hjk
n
i, j ∈ {1, . . . , n} and α ∈ {n + 1, . . . , n + p}, i.e. M isPtotally geodesic.
P
α
If λ2 = 1, then λ = 1 or λ = −1. If λ = −1, then α H α hα
= − k,α hα
ij
ik hjk ,
P
P
α α
so we have that H 2 + |h|2 = 0, i.e. h = 0. If λ = 1, then α H α hα
ij =
k,α hik hjk .
From equation (12), we have the following equation:
then
ij
P
(24)
Sij = c(n − 1)δij .
Remark 3.3. When p = 1, i.e. M
given in [5].
n
is a space-like hypersurface, the inequality
Corollary 3.4. If M n is a maximal space-like submanifold of indefinite space form
Mpn+p (c)(c > 0), S and ρ are Ricci curvature tensor and the scalar curvature of
M n , respectively, then
(25)
|S|2 = 2cρ(n − 1) − c2 n(n − 1)2
if and only if M n is totally geodesic.
Proof. If M n is totally geodesic, then from equations (12) and (13),
|S|2 = c2 n(n − 1)2 ,
and ρ = cn(n − 1) ,
i.e.
|S|2 = 2cρ(n − 1) − c2 n(n − 1)2 .
Conversely, from equations H = 0, (19), (20) and (21), we know that M n is
totally geodesic.
Theorem 3.5. If M n is a complete spacelike submanifold with flat normal bundle
and with positive sectional curvature immersed in indefinite space form Mpn+p (c),
(c > 0, p ≥ 2, n ≥ 2), S and ρ are Ricci curvature tensor and the scalar curvature
of M n , respectively, then
(26)
|S|2 = 2cρ(n − 1) − c2 n(n − 1)2
if and only if M n is totally geodesic.
Proof. If M n is totally geodesic, then from equations (12) and (13),
|S|2 = c2 n(n − 1)2 ,
and ρ = cn(n − 1) ,
i.e.
|S|2 = 2cρ(n − 1) − c2 n(n − 1)2 .
SPACELIKE SUBMANIFOLDS IN INDEFINITE SPACE FORM Mpn+p (c)
85
Conversely, from case (I) and case (II) in the proof of Theorem 3.2, we will prove
that M n must be geodesic under the conditions: λ = 1 and
X
X
α
(27)
H α hα
hα
ij =
ik hjk ,
α
k,α
for i, j ∈ {1, . . . , n}.
If H = 0, from Corollary 3.4, we know that M n is totally geodesic. Now we
H
suppose H 6= 0, and choose en+1 = |H|
. Then, it follows that
X
X
(28)
H=
hn+1
and H α =
hα
α > n + 1.
ii = 0 ,
ii en+1 ,
i
i
Since the normal bundle of M n is flat, we choose e1 , · · · , en such that
α
hα
ij = λi δij ,
for α = n + 1, . . . , n + p .
From equation (27), we have the following equations:
|H|2 = |H n+1 |2 = |h|2 .
(29)
Taking the covariant derivative of (29), we obtain
X
α
(30)
H n+1 Hkn+1 =
hα
ij hij,k
ijα
and by Lemma 3.1, we have
|H|2 |∇H|2 ≤ |h|2 |∇h|2 .
(31)
Then the Laplacian of |h|2 is given by:
X
1
1
α
4|h|2 = 4|H|2 = |∇h|2 +
hα
ij 4hij
2
2
ijα
X
1
2
n+1
α 2
= |∇h| +
λ
(H n+1 ) + Rijij (λα
i − λj )
2
i
(32)
We define an operator 2 acting on any function f by:
X
2f =
(H n+1 δij − hn+1
ij )f,ij
ij
Since (H
is trace free, it follows from [2] that 2 is self-adjoint relative
to L2 -inner product of M n , i.e.,
Z
Z
f 2g =
g2f .
n+1
δij − hn+1
ij )
Mn
Mn
Thus we have
2H n+1 =
X
n+1
(H n+1 δij − hn+1
ij )Hij
ij
(33)
X
1
λn+1 (H n+1 )
= 4|H|2 − |∇H|2 −
2
i
86
YINGBO HAN
From equations (30),(31),(32),(33),
1
α 2
(34)
2H n+1 ≥ Rijij (λα
i − λj ) .
2
Because Sij = c(n−1)δij , we see by the Bonnet-Myers theorem that M n is bounded
and hence compact.
Since 2 is self-adjoint, we have
Z
1
α 2
(35)
0≥
Rijij (λα
i − λj ) .
2
n
M
α
Then, by hypothesis Rijij > 0, so λα
i = λj for α ∈ {n + 1, . . . , n + p} and
i, j ∈ {1, . . . , n}.
From equation (27), we have
(36)
(n − 1)(λn+1
)2 = (λn+2
)2 + · · · + (λn+p
)2 .
1
1
1
From equation (28), we have
(37)
nλn+2
= · · · = nλn+p
= 0,
1
1
then we have
(38)
(n − 1)(λn+1
)2 = 0,
1
so λn+1
= λn+2
= · · · = λn+p
= 0, i.e. M n is a totally geodesic submanifold.
1
1
1
Acknowledgement. The author would like to thank Professors Y. X. Dong and
X. L. Chao for their valuable suggestions and useful discussions. Also, he would
like to thank the referee whose valuable suggestions make this paper more perfect.
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College of Mathematics and Information Science
Xinyang Normal University
Xinyang 464000, Henan, P. R. China
E-mail: yingbhan@yahoo.com.cn