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## Revista de la Unión Matemática Argentina

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*versión impresa* ISSN 0041-6932*versión 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

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

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

Next, let us prove that Since is parallel and the equality holds in (4.6) and (4.7), we arrive to

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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[4] A. Brasil, R.M.B. Chaves and G. Colares, *Rigidity results for submanifolds with parallel mean* *curvature vector in De Sitter space*, Glasg. Math. J. 48 (2006), 1-10. [ Links ]

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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*