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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006
On Complete Spacelike Submanifolds in the De Sitter Space With Parallel Mean Curvature Vector
Rosa Maria S. Barreiro Chaves and Luiz Amancio M. Sousa Jr.
Abstract: The text surveys some results concerning submanifolds with parallel mean curvature vector immersed in the De Sitter space. We also propose a semi-Riemannian version of an important inequality obtained by Simons in the Riemannian case and apply it in order to obtain some results characterizing umbilical submanifolds and a product of submanifolds in the (n + p)-dimensional De Sitter space .
2000 Mathematics Subject Classification. Primary 53C42, 53A10
Key words and phrases.De Sitter space, Simons type formula, complete spacelike submanifolds, parallel mean curvature vector
Let be an
-dimensional real vector space endowed with an inner product of index
given by
![x = (x1,x2,⋅⋅⋅ ,xn+p+1)](/img/revistas/ruma/v47n1/1a096x.png)
![ℝnp+p+1 .](/img/revistas/ruma/v47n1/1a097x.png)
We also define the semi-Riemannian manifold , by
.
is called
-dimensional De Sitter space of index
.
Let be an
-dimensional semi-Riemannian manifold immersed in
.
is said to be spacelike if the induced metric on
from the metric of
is positive definite.
From now on, we will consider spacelike submanifolds of
with parallel mean curvature vector
. Let
be the mean curvature of
. If
is parallel it is easy to verify that
is constant and, when
, these two conditions are equivalent. We say that
is a maximal submanifold if
vanishes identically.
It was proved by E. Calabi [6] (for ) and by S.Y. Cheng and S.T. Yau [8] (for all n) that a complete maximal spacelike hypersurface in
is totally geodesic. In [17], S. Nishikawa obtained similar results for others Lorentzian manifolds. In particular, he proved that a complete maximal spacelike hypersurface in
is totally geodesic. We recall that a submanifold
is said totally geodesic if its second fundamental form
vanishes identically.
A. Goddard [11] conjectured that the complete spacelike hypersurfaces of with
constant must be totally umbilical. The totally umbilical hypersurfaces of
are obtained by intersecting
with linear hyperplanes through the origin of
, where
can be viewed as hypersphere of
.
J. Ramanathan [19] proved Goddard's conjecture for and
. Moreover, if
he showed that the conjecture is false as can be seen from an example due to Dajczer-Nomizu [10]. In his proof, Ramanthan used the complex structure of
. K. Akutagawa [2] proved that Goddard's conjecture is true when
and
or when
and
. He also constructed complete spacelike rotation surfaces in
with constant
satisfying
and which are not totally umbilical.
In [15], S. Montiel proved that Goddard's conjecture is true provided that is compact. Furthermore, he exhibited examples of complete spacelike hypersurfaces with constant
satisfying
and being not totally umbilical - the so called hyperbolic cylinders (cf. [2] and [13]), which are isometric to the Riemannian product
of a hyperbolic line and an
-dimensional sphere of constant seccional curvatures
and
, respectively. Later, Montiel [16] studied complete spacelike hypersurfaces with constant mean curvature
and proved the following result.
Theorem 1.1. Let be a complete spacelike hypersurfaces in
with constant mean curvature
. If
is not connected at infinity, that is, if
has at least two ends, then
is, up to isometry, a hyperbolic cylinder.
Concerning to submanifolds of
with parallel mean curvature vector we may cite the following remarkable results. In [12], T. Ishihara proved the following theorem that generalizes for higher codimension the result of Cheng-Yau [8]
Theorem 1.2. Let be an n-dimensional complete Riemannian manifold isometrically immersed in
or
. If
is maximal, then the immersion is totally geodesic and
is a Riemannian space of constant curvature.
In [7], Q.M. Cheng showed that Akutagawa's result [2] is valid for higher codimensional complete spacelike submanifolds in with parallel mean curvature vector. More precisely, he proved the following result.
Theorem 1.3. Let be an n-dimensional complete spacelike submanifold in
with parallel mean curvature vector. If
, when n=2 or
, when
then
is totally umbilical.
In [14], H. Li obtained the following extension of Theorem 1.1.
Theorem 1.4. Let be an n-dimensional complete spacelike submanifold in
with parallel mean curvature vector. If
and
is not connected at infinity, that is, if
has at least two ends, then
is, up to isometry, a hyperbolic cylinder in
.
R. Aiyama [1] studied compact spacelike submanifold in with parallel mean curvature vector and proved the following results:
Theorem 1.5. Let be an n-dimensional compact spacelike submanifold in
with parallel mean curvature vector. If the normal connection of
is flat, then
is totally umbilical.
Theorem 1.6. Let be an n-dimensional compact spacelike submanifold in
with parallel mean curvature vector. If the sectional curvature of
is non-negative, then
is totally umbilical.
We point out that L. Alias and A. Romero [3] also obtained results related to complete spacelike submanifolds in with parallel mean curvature vector.
Let be an n-dimensional sphere in
with radius
and let
be an
-dimensional submanifold minimally immersed in
. Denote by
the second fundamental form of this immersion and by
the square of the length of
. In his pioneering work, J. Simons [20] proved the following inequality for
![]() | (1.1) |
As an application of formula (1.1), Simons [20] obtained the following result.
Theorem 1.7. Let be a closed minimal submanifold of
. Then either
is totally geodesic, or
, or
.
Two years later, S.S. Chern, M. do Carmo and S. Kobayashi [9], determined all the minimal submanifolds of satisfying
. More precisely, they proved:
Theorem 1.8. Let be a closed minimal submanifold of
. Assume that
. Then:
(i) Either (and
is totally geodesic) or
.
(ii) if and only if:
a) and
is locally a Clifford torus
.
b) and
is locally a Veronese surface in
.
In the case of a submanifold of
with non-zero parallel mean curvature vector
, it is convenient to modify slightly the second fundamental form
and to introduce the tracelless tensor
, where
is the mean curvature and g stands for the induced metric on
. W. Santos [21] established the following inequality for the Laplacian of
Let be a complete spacelike maximal submanifold of
. In [12], T. Ishihara derived the following inequality for
![]() | (1.2) |
As an important application of (1.2), Ishihara proved Theorem 1.2.
If is a spacelike hypersurface of
with constant mean curvature
, as in the Riemannian case, it is convenient to consider the tensor
. U.H. Ki, H.J. Kim and H. Nakagawa [13], established the following inequality for
![]() | (1.3) |
By applying (1.3) they obtained a constant that depends on
and
and such that
. They also characterized the hyperbolic cylinders as the only complete spacelike hypersurfaces of
with non-zero constant
and
. Moreover, they proved that a complete spacelike hypersurface of
with non-zero constant
and non-negative sectional curvature is totally umbilical, provided that
.
A. Brasil, G. Colares and O. Palmas [5] obtained the following gap theorem.
Theorem 1.9. Let ,
, be a complete spacelike hypersurface in
with constant mean curvature
. Then
and
a)either and
is totally umbilical or
b), where
are the roots of the polynomial
Recently, A. Brasil, R.M.B. Chaves and G. Colares [4] extended the above result for complete spacelike submanifolds in with parallel mean curvature vector.
Let be a spacelike submanifold of
with non-zero parallel mean curvature vector
and let
. Define the second fundamental form with respect to the normal direction
by
. If
denotes the squared norm of
, set
. In [7], Q. M. Cheng proved that
![]() | (1.4) |
Now we are going to state our main results. Theorem 1.10 is a Simons' type inequality for submanifolds in De Sitter space .
Theorem 1.10. Let be a spacelike submanifold immersed in
with parallel mean curvature. Then the following inequality holds
![]() | (1.5) |
Next Theorem is a Lorentzian version of results obtained by K. Yano and S. Ishihara [22] and also by S.T. Yau [23] for Riemannian submanifolds.
Theorem 1.11. Let be a complete spacelike submanifold in
with parallel mean curvature vector and non-negative sectional curvature. If
has constant scalar curvature R, then
is totally umbilical or a product
, where each
is a totally umbilical submanifold of
and the
are mutually perpendicular along their intersections.
As we saw in the Theorem 1.6, compact spacelike submanifolds in with parallel mean curvature vector and non-negative sectional curvature are totally umbilic.
The following result is an application of formula (1.5).
Theorem 1.12. Let be a complete spacelike submanifold in
with parallel mean curvature vector. If
denotes the function that assigns to each point of
the supremum of the sectional curvatures at that point, there exists a constant
such that if
, then either:
(i) and
is totally umbilical or
(ii) and
is totally geodesic.
In this section we will introduce some basic facts and notations that will appear on the paper. Let be an
-dimensional Riemannian manifold immersed in
. As the indefinite Riemannian metric of
induces the Riemannian metric of
, the immersion is called spacelike. We choose a local field of semi-Riemannian orthonormal frames
in
such that, at each point of
,
span the tangent space of
. We make the following standard convention of indices
Take the correspondent dual coframe such that the semi-Riemannian metric of
is given by
Then the structure equations of
are given by
![]() | (2.1) |
![]() | (2.2) |
![]() | (2.3) |
Next, we restrict those forms to . First of all we get
![]() | (2.4) |
So the Riemannian metric of is written as
.
Since from Cartan's lemma, we can write
![]() | (2.5) |
Set ,
and
the second fundamental form, the mean curvature vector and the mean curvature of
, respectively.
Using the structure equations we obtain the Gauss equation
![]() | (2.6) |
The scalar curvature is given by
![]() | (2.7) |
where is the squared norm of the second fundamental form of
.
We also have the structure equations of the normal bundle of
![]() | (2.8) |
![]() | (2.9) |
where
![]() | (2.10) |
The covariant derivatives of
satisfy
![]() | (2.11) |
Then, by exterior differentiation of (2.5), we obtain the Codazzi equation
![]() | (2.12) |
Similarly, we have the second covariant derivatives of
so that
![]() | (2.13) |
By exterior differentiation of (2.11), we can get the following Ricci formula
![]() | (2.14) |
The Laplacian of
is defined by
. From (2.12) and (2.14), we have
![]() | (2.15) |
If , we choose
Thus
![]() | (2.16) |
where denotes the matrix
From (2.6), (2.10), (2.15) and (2.16) it is straightforward to verify that
![]() | (2.17) |
where , for all matrix
Recall that is a submanifold with parallel mean curvature vector
if
where
is the normal connection of
in
Note that this condition implies that
is constant and
![]() | (2.18) |
We will need the following generalized Maximum Principle due to Omori and Yau (cf. [18] and [23]).
Lemma 2.1. Let be a complete Riemannian manifold with Ricci curvature bounded from below and let
be a
-function which is bounded from below on
. Then there is a sequence of points
in
such that
and
We also will need the following algebraic Lemma (for a proof see [21]).
Lemma 2.2. Let be symmetric linear maps such that
and
Then
![]() | (2.19) |
and the equality holds if and only if of the eigenvalues
of
and the corresponding eigenvalues
of
satisfy
![]() | (2.20) |
3. Proof of Simons' type Inequality
Proof of Theorem 1.10. If , set
and consider the following symmetric tensor
![]() | (3.1) |
It is easy to check that is traceless and
![]() | (3.2) |
where denotes the matrix
.
Because is parallel, we have
constant. Moreover, as
, we can choose a local field of orthonormal frames
such that
. With this choice (2.16) implies that
![]() | (3.3) |
![]() | (3.4) |
Since is parallel, from (2.17), (3.2), (3.3) and (3.4) we have
![]() | (3.5) |
As the matrices and
are traceless and the matrix
comutes with all the matrices
, we can apply Lemma 2.2 in order to obtain
![]() | (3.6) |
Due to Cauchy-Schwarz inequality we can write
![]() | (3.7) |
It follows from (3.5), (3.6) and (3.7) that formula (1.5) holds.
If ,
is said to be maximal. In this case, from (1.2) we have
![]() | (3.8) |
4. Proofs of Theorems 1.11 and 1.12
Proof of Theorem 1.11. Since the mean curvature vector is parallel and
, from (2.15) we have
![]() | (4.1) |
Next, we will obtain a pointwise estimate for the last two terms. For each fixed let
be an eigenvalue of
, i.e.
and denote by
the infimum of the sectional curvatures at a point
of
. Then
![]() | (4.2) |
It implies that
![]() | (4.3) |
As parallel implies
constant, by (2.7) we see that
is also constant, thus
.
Since , from (4.1) and (4.3), we get
![]() | (4.4) |
It turns out that:
i) , for all
and
and so the normal bundle of
is flat. Hence, all the matrices
can be diagonalized simultaneously;
ii) and so the second fundamental form
is parallel. In particular, it implies that
is constant for all
.
From i), ii), (4.1) and (4.2) we can write and, since
, we obtain
.
Consequentely, we may apply the same methods used by Ishihara (see [12], Lemmas 5.1, 5.2 and Theorem 1.3) to conclude that is totally umbilical or a product
where
is a totally umbilical submanifold in
and the
are mutually perpendicular along their intersections.
Remark: Let be a complete spacelike submanifold in
with parallel mean curvature vector and non-negative sectional curvature. In (4.4), we got the inequality
, which shows that
is a subharmonic smooth function. Therefore, if the supremum of
is attained on
, it follows from the Maximum Principle that S is constant and we have the same conclusions as in Theorem 1.11.
Proof of Theorem 1.12. In the proof of Theorem 1.10 we used the following inequality
![]() | (4.5) |
Applying the same arguments as in the proof of the inequality (4.3), we obtain
![]() | (4.6) |
For technical reasons, we will write the expression (4.1) for the Laplacian of as
![]() | (4.7) |
Thus, from (4.5), (4.6) and (4.7), if , we have
![]() | (4.8) |
Using similar arguments as in [14], it is possible to show that . Therefore, we can apply Lemma 2.1 to the function
and obtain a sequence of points
in
such that
![]() | (4.9) |
By applying inequality (4.8) at , taking the limit, and using (4.9) we get
![]() | (4.10) |
If , it can be easily checked that
![sup K = β(n, p,H)](/img/revistas/ruma/v47n1/1a09367x.png)
![--pn(n---2)-- sup ∣ Φ ∣= 2 ∘n-(n----1)](/img/revistas/ruma/v47n1/1a09368x.png)
Thus, if , from (4.10) and the last inequality we conclude that
and
is totally umbilical.
If , we will suppose that
is not totally umbilical and derive a contradiction. First, let us prove that
. Notice that
It shows that all the estimates used to obtain inequality (4.10) turn into equalities. More precisely, (3.6) and (3.7) can now be written as
![]() | (4.11) |
![]() | (4.12) |
As mentioned before, taking subsequences if necessary, we can arrive to a sequence in
, which satisfies (4.9) and such that
![]() | (4.13) |
By evaluating (4.11) at , taking the limit for
and using (4.13) it gives
![]() | (4.14) |
Since we have
![]() | (4.15) |
Hence, By evaluating (4.12) at
and taking the limit for
, from (4.13) and (4.15), we get
![p = 1](/img/revistas/ruma/v47n1/1a09390x.png)
Next, let us prove that Since
is parallel and the equality holds in (4.6) and (4.7), we arrive to
![sup K = 0](/img/revistas/ruma/v47n1/1a09394x.png)
Now we are in position to prove that is totally umbilical. Observe that
and
yield
Hence . In this case, according to Montiel (cf. [16], Proposition 2), either
is a totally umbilical hypersurface or
and the supremum of the scalar curvature of
is equal to
.
As is not totally umbilical, we conclude that the supremum of the scalar curvature of
is equal to
, which contradicts the fact that
. Therefore,
is totally umbilical.
Because is arbitrary, taking the limit for
in
, we get
.
Moreover, since is totally umbilical, if
we obtain
thus
, which implies
and shows that
is totally geodesic.
Acknowledgements. The authors would like to express their thanks to Fernanda Ester C. Camargo for valuable comments and suggestions about this paper, as well as to the referee for his careful reading of the original manuscript. This work was carried out while the second author was visiting the Institute of Mathematics and Statistics at the University of São Paulo (Brazil). He would like to thank Professor Claudio Gorodski and Professor Paolo Piccione for the warm hospitality and financial support, during his visit.
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Rosa Maria S. Barreiro Chaves
Instituto de Matemática e Estatística
Universidade de São Paulo, Rua do Matão, 1010,
São Paulo - SP, Brazil, CEP 05508-090
rosab@ime.usp.br
Luiz Amancio M. Sousa Jr.
Departamento de Matemática e Estatística
Universidade Federal do Estado do Rio de Janeiro, Avenida Pasteur, 458,
Urca, Rio de Janeiro - RJ, Brazil, CEP 22290-240
amancio@impa.br
Recibido: 17 de noviembre de 2005
Aceptado: 22 de septiembre de 2006