versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006
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
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  (for ) and by S.Y. Cheng and S.T. Yau  (for all n) that a complete maximal spacelike hypersurface in is totally geodesic. In , 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  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  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 . In his proof, Ramanthan used the complex structure of . K. Akutagawa  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 , 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.  and ), which are isometric to the Riemannian product of a hyperbolic line and an -dimensional sphere of constant seccional curvatures and , respectively. Later, Montiel  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 , T. Ishihara proved the following theorem that generalizes for higher codimension the result of Cheng-Yau 
In , Q.M. Cheng showed that Akutagawa's result  is valid for higher codimensional complete spacelike submanifolds in with parallel mean curvature vector. More precisely, he proved the following result.
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  studied compact spacelike submanifold in with parallel mean curvature vector and proved the following results:
We point out that L. Alias and A. Romero  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  proved the following inequality for
Two years later, S.S. Chern, M. do Carmo and S. Kobayashi , 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  established the following inequality for the Laplacian of
Let be a complete spacelike maximal submanifold of . In , T. Ishihara derived the following inequality for
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 , established the following inequality for
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  obtained the following gap theorem.
Recently, A. Brasil, R.M.B. Chaves and G. Colares  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 , Q. M. Cheng proved that
Now we are going to state our main results. Theorem 1.10 is a Simons' type inequality for submanifolds in De Sitter space .
Theorem 1.11. 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
Next, we restrict those forms to . First of all we get
So the Riemannian metric of is written as .
Since from Cartan's lemma, we can write
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
The scalar curvature is given by
where is the squared norm of the second fundamental form of .
We also have the structure equations of the normal bundle of
The covariant derivatives of satisfy
Then, by exterior differentiation of (2.5), we obtain the Codazzi equation
Similarly, we have the second covariant derivatives of so that
By exterior differentiation of (2.11), we can get the following Ricci formula
If , we choose Thus
where denotes the matrix
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
We also will need the following algebraic Lemma (for a proof see ).
and the equality holds if and only if of the eigenvalues of and the corresponding eigenvalues of satisfy
Proof of Theorem 1.10. If , set and consider the following symmetric tensor
It is easy to check that is traceless and
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
As the matrices and are traceless and the matrix comutes with all the matrices , we can apply Lemma 2.2 in order to obtain
Due to Cauchy-Schwarz inequality we can write
If , is said to be maximal. In this case, from (1.2) we have
It implies that
As parallel implies constant, by (2.7) we see that is also constant, thus .
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 .
Consequentely, we may apply the same methods used by Ishihara (see , 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.
Applying the same arguments as in the proof of the inequality (4.3), we obtain
For technical reasons, we will write the expression (4.1) for the Laplacian of as
If , it can be easily checked that
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
As mentioned before, taking subsequences if necessary, we can arrive to a sequence in , which satisfies (4.9) and such that
Since we have
Now we are in position to prove that is totally umbilical. Observe that and yield
Hence . In this case, according to Montiel (cf. , 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
, 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
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
Recibido: 17 de noviembre de 2005
Aceptado: 22 de septiembre de 2006