Characterizations of complete spacelike submanifolds in the $\mathbf{(n+p)}$-dimensional anti-de Sitter space of index $\mathbf{q}$

Abstract

Our purpose in this paper is to study the geometry of n-dimensional complete spacelike submanifolds immersed in the $(n+p)$-dimensional anti-de Sitter space $\mathbb {H}^{n+p}_{q}$ of index q, with $1\le q\le p$. Under suitable constraints on the Ricci curvature and the second fundamental form, we show that a complete maximal spacelike submanifold of $\mathbb {H}^{n+p}_{q}$ must be totally geodesic. Furthermore, we establish sufficient conditions to guarantee that a complete spacelike submanifold with nonzero parallel mean curvature vector in $\mathbb {H}^{n+p}_{p}$ must be pseudo-umbilical, which means that its mean curvature vector is an umbilical direction.

1 Introduction

Apart from their physical importance (see, for example, [25, 34]), the interest in the study of spacelike submanifolds immersed in a Lorentzian space is motivated by their nice Bernstein-type properties. For instance, it was proved by Calabi [10] (for $n\le 4$) and by Cheng and Yau [14] (for all n) that the only complete maximal spacelike hypersurfaces of the Lorentz-Minkowski space $\mathbb {L}^{n+1}$ are the spacelike hyperplanes. In [29], Nishikawa proved that a complete maximal spacelike hypersurface (that is, with mean curvature identically zero) in the de Sitter space $\mathbb {S}_{1}^{n+1}$ must be totally geodesic. In [18], Goddard conjectured that the complete spacelike hypersurfaces of $\mathbb {S}_{1}^{n+1}$ with constant mean curvature H must be totally umbilical. Ramanathan [32] proved Goddard’s conjecture in $\mathbb {S}_{1}^{3}$ for $0\le H\le 1$. Moreover, for $H>1$, he showed that the conjecture is false, as can be seen from an example due to Dajczer and Nomizu in [16]. Independently, Akutagawa [2] proved that Goddard’s conjecture is true when either $n=2$ and $H^{2}\le 1$ or $n\ge 3$ and $H^{2}<\frac{4(n-1)}{n^{2}}$. He also constructed complete spacelike rotation surfaces in $\mathbb {S}_{1}^{3}$ having constant mean curvature $H>1$ and which are not totally umbilical. Next, Montiel [26] showed that Goddard’s conjecture is true provided that $M^{n}$ is compact. Furthermore, he exhibited examples of complete spacelike hypersurfaces in $\mathbb {S}_{1}^{n+1}$ with constant mean curvature H satisfying $H^{2}\ge \frac{4(n-1)}{n^2}$ and being non totally umbilical, the so-called hyperbolic cylinders.

In higher codimension, Cheng [12] extended Akutagawa’s result for complete spacelike submanifolds with parallel mean curvature vector (that is, the mean curvature vector field is parallel as a section of the normal bundle) in the de Sitter space $\mathbb {S}_{p}^{n+p}$ of index p. Afterwards, Aiyama [1] studied compact spacelike submanifolds in $\mathbb {S}_{p}^{n+p}$ with parallel mean curvature vector and proved that if the normal connection of $M^{n}$ is flat, then $M^{n}$ is totally umbilical. Furthermore, she proved that a compact spacelike submanifold in $\mathbb {S}_{p}^{n+p}$ with parallel mean curvature vector and nonnegative sectional curvature must be totally umbilical. Meanwhile, Alías and Romero [4] developed some integral formulas for compact spacelike submanifolds in $\mathbb {S}_{p}^{n+p}$ which have a very clear geometric meaning and, as application, they obtained a Bernstein type result for complete maximal submanifolds in $\mathbb {S}_{q}^{n+p}$, extending a previous result due to Ishihara [19]. Moreover, they extended Ramanathan’s result [32] showing that the only compact spacelike surfaces in $\mathbb {S}_{p}^{2+p}$ with parallel mean curvature vector are the totally umbilical ones and, in particular, they also reproved Cheng’s result [14] establishing that every complete spacelike surface in $\mathbb {S}_{p}^{2+p}$ with parallel mean curvature vector such that $H^2<1$ is totally umbilical. Next, Li [22] showed that Montiel’s result [26] still holds for higher codimensional spacelike submanifolds in $\mathbb {S}_{p}^{n+p}$. More recently, Araújo and Barbosa [6], assuming appropriated controls on the second fundamental form and on the scalar curvature, extended the techniques developed in [23, 33, 35] and proved that a compact spacelike submanifold in $\mathbb {S}_{p}^{n+p}$ with nonzero mean curvature and parallel mean curvature vector must be isometric to a sphere.

When the ambient spacetime is the anti-de Sitter space $\mathbb H_1^{n+1}$, Choi et al. [15] used the generalized maximum principle of Omori [30] and Yau [36] in order to obtain a Myers type theorem [28] concerning complete maximal spacelike hypersurfaces. More precisely, they showed that if the height function with respect to a timelike vector of such a hypersurface obeys a certain boundedness, then it must be totally geodesic. Extending a technique due to Yau [37], the first author jointly with Camargo [7] obtained another rigidity results to complete maximal spacelike hypersurfaces in $\mathbb H_1^{n+1}$, imposing suitable conditions on both the norm of the second fundamental form and a certain height function naturally attached to the hypersurface. Afterwards, working with a suitable warped product model for an open subset of $\mathbb H_1^{n+1}$, the same authors jointly with Caminha and Parente [8] extended the main result of [7] showing that if $M^n$ is a complete spacelike hypersurface with constant mean curvature and bounded scalar curvature in $\mathbb H_1^{n+1}$, such that the gradient of its height function with respect to a timelike vector has integrable norm, then $M^n$ must be totally umbilical. More recently, the first author jointly Aquino [5] obtained another characterizations theorems concerning complete constant mean curvature spacelike hypersurfaces of $\mathbb H_1^{n+1}$, under suitable constraints on the behavior of the Gauss mapping. In higher codimension, Ishihara [19] proved that a n-dimensional complete maximal spacelike submanifold immersed in the anti-de Sitter space $\mathbb H_p^{n+p}$ of index p must have the squared norm of the second fundamental form bounded from above by np. Moreover, the only ones that attain this estimate are the hyperbolic cylinders. Later on, Cheng [13] obtained a refinement of Ishihara’s result [19] for the case of complete maximal spacelike surfaces immersed in $\mathbb H_p^{2+p}$.

Motivated by the works above described, our purpose in this paper is to study the geometry of complete spacelike submanifolds immersed in the anti-de Sitter space $\mathbb {H}^{n+p}_{q}$ of index q. In this setting, we extend the technique due to Alías and Romero in [4] and, under appropriated constraints on the Ricci curvature and second fundamental form, we show that a complete maximal spacelike submanifold $M^n$ of $\mathbb {H}^{n+p}_{q}$ must be totally geodesic (see Theorem 1 and Corollaries 1 and 2). Furthermore, we establish sufficient conditions to guarantee that a complete spacelike submanifold with nonzero parallel mean curvature vector $\mathbf{H}$ in $\mathbb {H}^{n+p}_{p}$ must be pseudo-umbilical, which means that $\mathbf{H}$ is an umbilical direction (see Theorem 2 and Corollary 3). Our approach is based on a generalized form of a maximum principle at the infinity of Yau [37] (see Lemma 1 and Remark 1).

2 Preliminaries

Let $\mathbb {R}^{n+p+1}_{q+1}$ be the $(n+p+1)$-dimensional semi-Euclidean space endowed with metric tensor $\langle ,\,\rangle$ of index q, with $1\le q\le p$, given by

$$\begin{aligned} \langle v,w\rangle =\sum _{i=1}^{n+p-q}\!\!\!\!\!v_{i}w_{i}-\!\!\!\!\!\!\sum _{j=n+p-q+1}^{n+p+1}\!\!\!\!\!\!v_{j}w_{j}, \end{aligned}$$

and let $\mathbb {H}^{n+p}_{q}$ be the $(n+p)$-dimensional unitary anti-de Sitter space of index q, that is,

$$\begin{aligned} \mathbb {H}^{n+p}_{q}=\{x\in \mathbb {R}^{n+p+1}_{q+1}\,;\,\langle x,x\rangle =-1\}, \end{aligned}$$

which has constant sectional curvature equal to $-1$.

Along this work, we will consider $x:M^{n}\rightarrow \mathbb {H}^{n+p}_{q}\subset \mathbb {R}^{n+p+1}_{q+1}$ a spacelike submanifold isometrically immersed in $\mathbb {H}^{n+p}_{q}$. We recall that a submanifold immersed is said to be spacelike if its induced metric is positive definite. In this setting, we will denote by $\nabla ^{\circ }$, $\overline{\nabla }$ and $\nabla$ the Levi-Civita connections of $\mathbb {R}^{n+p+1}_{q+1}$, $\mathbb {H}^{n+p}_{q}$ and $M^{n}$, respectively, and $\nabla ^{\perp }$ will stand for the normal connection of $M^{n}$ in $\mathbb {H}^{n+p}_{q}$.

We will denote by $\alpha$ the second fundamental form of $M^{n}$ in $\mathbb {H}^{n+p}_{q}$ and by $A_{\xi }$ the shape operator associated to a fixed vector field $\xi$ normal to $M^{n}$ in $\mathbb {H}^{n+p}_{q}$. We note that, for each $\xi \in \mathfrak {X}^{\perp }(M)$, $A_{\xi }$ is a symmetric endomorphism of the tangent space $T_{x}M$ at $x\in M^{n}$. Moreover, $A_{\xi }$ and $\alpha$ are related by

$$\begin{aligned} \langle A_{\xi }X,Y\rangle =\langle \alpha (X,Y),\xi \rangle , \end{aligned}$$

(2.1)

for all tangent vector fields $X,Y\in \mathfrak {X}(M)$.

We also recall that the Gauss and Weingarten formulas of $M^{n}$ in $\mathbb {H}^{n+p}_{q}$ are given by

$$\begin{aligned} \nabla ^{\circ }_{X}Y=\overline{\nabla }_{X}Y+\langle X,Y\rangle x=\nabla _{X}Y+\alpha (X,Y)+\langle X,Y\rangle x, \end{aligned}$$

(2.2)

and

$$\begin{aligned} \nabla ^{\circ }_{X}\xi =\overline{\nabla }_{X}\xi =-A_{\xi }X+\nabla ^{\perp }_{X}\xi , \end{aligned}$$

(2.3)

for all tangent vector fields $X,Y\in \mathfrak {X}(M)$ and normal vector field $\xi \in \mathfrak {X}^{\perp }(M)$.

As in [31], the curvature tensor R of the spacelike submanifold $M^{n}$ is given by

$$\begin{aligned} R(X,Y)Z=\nabla _{[X,Y]}Z-[\nabla _{X},\nabla _{Y}]Z, \end{aligned}$$

where [, ] denotes the Lie bracket and $X,Y,Z\in \mathfrak {X}(M)$.

A well known fact is that the curvature tensor R of $M^{n}$ can be described in terms of its second fundamental form $\alpha$ and the curvature tensor $\overline{R}$ of the ambient spacetime $\mathbb {H}^{n+p}_{q}$ by the so-called Gauss equation, which is given by

$$\begin{aligned} \langle R(X,Y)Z,W\rangle= & {} \langle Y,Z\rangle \langle X,W\rangle -\langle X,Z\rangle \langle Y,W\rangle \nonumber \\&+\,\langle \alpha (X,W),\alpha (Y,Z)\rangle -\langle \alpha (X,Z),\alpha (Y,W)\rangle , \end{aligned}$$

(2.4)

for all tangent vector fields $X,Y,Z,W\in \mathfrak {X}(M)$. Moreover, Codazzi equation asserts that

$$\begin{aligned} (\nabla _XA_\xi )Y=(\nabla _YA_\xi )X, \end{aligned}$$

(2.5)

for all $X,Y\in \mathfrak {X}(M)$ and $\xi \in \mathfrak {X}^\perp (M)$.

We will define the mean curvature vector of $M^{n}$ in $\mathbb {H}^{n+p}_{q}$ by

$$\begin{aligned} \mathbf{H}=\frac{1}{n}\mathrm{tr}(\alpha ). \end{aligned}$$

We recall that $M^{n}$ is called maximal when $\mathbf{H}\equiv 0$ and we say that $M^{n}$ has parallel mean curvature vector when $\nabla ^{\perp }_{X}{} \mathbf{H}\equiv 0$, for every $X\in \mathfrak {X}(M)$. In this last case, when $q=p$ and $\mathbf{H}\ne 0$, we have that $\langle \mathbf{H},\mathbf{H}\rangle$ is a negative constant along $M^{n}$. Moreover, $M^{n}$ is called totally geodesic when its second fundamental form $\alpha$ vanishes identically and it is called totally umbilical when

$$\begin{aligned} \alpha (X,Y)=\langle X,Y\rangle \mathbf{H}, \end{aligned}$$

(2.6)

for all tangent vector fields $X,Y\in \mathfrak {X}(M)$.

We close this section describing the main analytical tool which is used along the proofs of our results in the next sections. In [37] Yau, generalizing a previous result due to Gaffney [17], established the following version of Stokes’ Theorem on an n-dimensional, complete noncompact Riemannian manifold $M^n$: if $\omega \in \Omega ^{n-1}(M)$ is an integrable $(n-1)$-differential form on $M^n$, then there exists a sequence $B_i$ of domains on $M^n$ such that $B_i\subset B_{i+1}$, $M^n=\bigcup _{i\ge 1}B_i$ and

$$\begin{aligned} \lim _{i\rightarrow +\infty }\displaystyle \int _{B_i}d\omega =0. \end{aligned}$$

Suppose that $M^n$ is oriented by the volume element dM. If $\omega =\iota _XdM$ is the contraction of dM in the direction of a smooth vector field X on $M^n$, then Caminha obtained a suitable consequence of Yau’s result, which can be regarded as an extension of Hopf’s maximum principle for complete Riemannian manifolds (cf. Proposition 2.1 of [9]). In what follows, $\mathcal {L}^1(M)$ and $\mathrm{div}$ denote the space of Lebesgue integrable functions and the divergence on $M^{n}$, respectively.

Lemma 1

Let X be a smooth vector field on the n-dimensional complete noncompact oriented Riemannian manifold $M^{n}$, such that $\mathrm{div}X$ does not change sign on $M^{n}$. If $|X|\in \mathcal {L}^{1}(M)$, then $\mathrm{div}X=0$.

Remark 1

Lemma 1 can also be seen as a consequence of the version of Stokes’ Theorem given by Karp in [20]. In fact, using Theorem in [20], condition $|X|\in \mathcal L^1(M)$ can be weakened to the following technical condition:

$$\begin{aligned} \liminf _{r\rightarrow +\infty }\frac{1}{r}\int _{B(2r)\setminus B(r)}|X|dM=0, \end{aligned}$$

where B(r) denotes the geodesic ball of radius r center at some fixed origin $o\in M^n$. See also Corollary 1 and Remark in [20] for some another geometric conditions guaranteing this fact.

Remark 2

Reasoning in a similar way of that in the beginning of Section 4 of [5] (see also Section 4 in [3]), it is not difficult to verify that there exist no n-dimensional compact (without boundary) spacelike submanifolds immersed in $\mathbb {H}^{n+p}_{p}$. Motivated by this fact, along this paper we will deal with complete spacelike submanifolds.

3 Complete maximal submanifolds immersed in $\mathbb {H}^{n+p}_{q}$

Let $a\in \mathbb {R}^{n+p+1}_{q+1}$ be a fixed arbitrary vector and put

$$\begin{aligned} a=a^{\top }+a^{N}-\langle a,x\rangle x, \end{aligned}$$

(3.1)

where $a^{\top }\in \mathfrak {X}(M)$ and $a^{N}\in \mathfrak {X}^{\perp }(M)$ denote, respectively, the tangential and normal components of a with respect to $M^{n}\hookrightarrow \mathbb {H}^{n+p}_{q}$. By taking covariant derivative in (3.1) and using (2.2) and (2.3), we get for all tangent vector field $X\in \mathfrak {X}(M)$ that

$$\begin{aligned} \nabla _{X}a^{\top }=A_{a^{N}}X+\langle a,x\rangle X \end{aligned}$$

(3.2)

and

$$\begin{aligned} \nabla _{X}^{\perp }a^{N}=-\alpha (a^{\top },X). \end{aligned}$$

(3.3)

Hence, from (2.1) and (3.2) we obtain

$$\begin{aligned} \mathrm{div}(a^{\top })=\mathrm{tr}(A_{a^{N}})+n\langle a,x\rangle =n\langle a,\mathbf{H}\rangle +n\langle a,x\rangle . \end{aligned}$$

(3.4)

Moreover, we also have that

$$\begin{aligned} \mathrm{tr}(\nabla _{a^{\top }}A_{\xi })= & {} \sum _{i}\langle \nabla _{a^{\top }}A_{\xi }e_{i},e_{i}\rangle -\sum _{i}\langle \nabla _{a^{\top }}e_{i},A_{\xi }e_{i}\rangle \\&+\,n\langle \nabla ^{\perp }_{a^{\top }}{} \mathbf{H},\xi \rangle -\sum _{i}a^{\top }\langle A_{\xi }e_{i},e_{i}\rangle .\nonumber \end{aligned}$$

So, considering a local orthonormal frame $\{e_{1},\ldots ,e_{n}\}$ on $M^{n}$ such that $A_{\xi }e_{i}=\lambda ^{\xi }_{i}e_{i}$, with a straightforward computation we can verify that

$$\begin{aligned} \mathrm{tr}(\nabla _{a^{\top }}A_{\xi })=n\langle \nabla ^{\perp }_{a^{\top }}{} \mathbf{H},\xi \rangle . \end{aligned}$$

(3.5)

From Codazzi Eq. (2.5) jointly with (3.2) and (3.5) we obtain, for all $\xi \in \mathfrak {X}^{\perp }(M)$,

$$\begin{aligned} \mathrm{div}(A_{\xi }a^{\top })= & {} n\langle \nabla _{a^{\top }}^{\perp }{} \mathbf{H},\xi \rangle +\mathrm{tr}(A_{a^{N}}\circ A_{\xi })+\langle a,x\rangle \mathrm{tr}(A_{\xi })\nonumber \\&+\sum _{i}\langle \alpha (a^{\top },e_{i}),\nabla _{e_{i}}^{\perp }\xi \rangle . \end{aligned}$$

(3.6)

On the other hand, taking the trace in Gauss Eq. (2.4), we have

$$\begin{aligned} \mathrm{Ric}(X,Y)=-(n-1)\langle X,Y\rangle +n\langle \alpha (X,Y),\mathbf{H}\rangle -\sum _{i}\langle \alpha (X,e_{i}),\alpha (Y,e_{i})\rangle , \end{aligned}$$

(3.7)

where $\mathrm{Ric}$ denotes the Ricci curvature of $M^{n}$. Considering $X=Y=a^{\top }$ in (3.7), we obtain

$$\begin{aligned} \mathrm{Ric}(a^{\top },a^{\top })= & {} -(n-1)\big |a^{\top }\big |^{2}+n\langle \alpha (a^{\top },a^{\top }),\mathbf{H}\rangle \nonumber \\&\!\!\!-\sum _{i}\langle \alpha (a^{\top },e_{i}),\alpha (a^{\top },e_{i})\rangle . \end{aligned}$$

(3.8)

Furthermore, from (3.3) and (3.6) we get

$$\begin{aligned} \mathrm{div}(A_{a^{N}}a^{\top })= & {} n\langle \nabla _{a^{\top }}^{\perp }{} \mathbf{H},a^{N}\rangle +\mathrm{tr}(A_{a^{N}}^{2})+\langle a,x\rangle \mathrm{tr}(A_{a^{N}})\nonumber \\&-\sum _{i}\langle \alpha (a^{\top },e_{i}),\alpha (a^{\top },e_{i})\rangle . \end{aligned}$$

(3.9)

Hence, from (3.8) and (3.9) we conclude that

$$\begin{aligned} \mathrm{div}(A_{a^{N}}a^{\top })= & {} n\langle \nabla _{a^{\top }}^{\perp }{} \mathbf{H},a^{N}\rangle +\mathrm{tr}(A_{a^{N}}^{2})+\langle a,x\rangle \mathrm{tr}(A_{a^{N}})\nonumber \\&+\,\mathrm{Ric}(a^{\top },a^{\top })+(n-1)\big |a^{\top }\big |^{2}-n\langle \alpha (a^{\top },a^{\top }),\mathbf{H}\rangle . \end{aligned}$$

(3.10)

Based on the previous computations, we obtain the following Bernstein type result concerning maximal submanifolds immersed in $\mathbb {H}^{n+p}_{q}$

Theorem 1

Let $M^{n}$ be a complete maximal spacelike submanifold immersed in $\mathbb {H}^{n+p}_{q}$, with $1\le q\le p$. Suppose that $Ric\ge -(n-1)$ on $M^{n}$. If there exist p vectors $a_1,\ldots ,a_p\in \mathbb {R}^{n+p+1}_{q+1}$ such that $a_1^N,\ldots ,a_p^N$ are linearly independent, with $A_{a_i^{N}}$ bounded on $M^{n}$ and $|a_i^{\top }|\in \mathcal {L}^{1}(M)$ for each $1\le i\le p$, then $M^{n}$ is totally geodesic.

Proof

Let us consider $a=a_i$ for some $i\in \{1,\dots ,p\}$. Provided that $\mathbf{H}=0$, from (2.1) we see that Eq. (3.10) can be rewritten as follows

$$\begin{aligned} \mathrm{div}(A_{a^{N}}a^{\top })=\mathrm{tr}(A_{a^{N}}^{2})+\mathrm{Ric}(a^{\top },a^{\top })+(n-1)\big |a^{\top }\big |^{2}. \end{aligned}$$

(3.11)

Thus, since $\mathrm{Ric}(a^{\top },a^{\top })\ge -(n-1)\big |a^{\top }\big |^{2}$, from (3.11) we obtain that

$$\begin{aligned} \mathrm{div}(A_{a^{N}}a^{\top })\ge 0. \end{aligned}$$

(3.12)

Moreover, whereas $A_{a^{N}}$ is bounded on $M^{n}$, $|A_{a^{N}}|\le C_{1}$, for some constant $C_{1}>0$. Thus, as we are assuming that $|a^{\top }|\in \mathcal {L}^{1}(M)$, we have

$$\begin{aligned} |A_{a^{N}}a^{\top }|\le |A_{a^{N}}||a^{\top }|\le C_{1}|a^{\top }|\in \mathcal {L}^{1}(M). \end{aligned}$$

(3.13)

Hence, taking into account (3.12) and (3.13), we can apply Lemma 1 to guarantee that $\mathrm{div}(A_{a^{N}}a^{\top })=0$. Consequently, returning to Eq. (3.11), we conclude that $A_{a^{N}}\equiv 0$. Therefore, since $\alpha (X,Y)=\sum _{i=1}^p\langle A_{a_i^{N}}X,Y\rangle a_i^{N}$, we have that $M^{n}$ must be totally geodesic. $\square$

Remark 3

Despite our assumption on the $a_{i}^{N}$ in Theorem 1 to be a technical hypothesis, it is motivated by the fact that it occurs in a natural way in the context of spacelike hypersurfaces (see Corollary 2). In this sense, it is a mild hypothesis.

In the case that $p=q$, being $M^{n}$ a maximal submanifold of $\mathbb {H}^{n+p}_{p}$, a classical result due to Ishihara [19] assures us that $|A|^{2}\le np$ (see also Cheng [13] for the case $n=2$). Moreover, each maximal submanifold in $\mathbb {H}^{n+p}_{p}$ meets the condition $\mathrm{Ric}\ge -(n-1)$. Thus, as a consequence of Theorem 1 we obtain

Corollary 1

Let $M^{n}$ be a complete maximal spacelike submanifold immersed in $\mathbb {H}^{n+p}_{p}$. If there exist p vectors $a_1,\ldots ,a_p\in \mathbb {R}^{n+p+1}_{p+1}$ such that $a_1^N,\ldots ,a_p^N$ are linearly independent, with $|a_i^{\top }|\in \mathcal {L}^{1}(M)$ for each $1\le i\le p$, then $M^{n}$ is totally geodesic.

Taking into account that the warped product model $-(-\frac{\pi }{2},\frac{\pi }{2})~\times _{\cos t}~\mathbb H^n$, which is considered in [7] in order to prove their results, models just only an open subset of $\mathbb {H}^{n+1}_{1}$ (cf. Example 4.3 of [27]), from Corollary 1 we obtain the following improvement of Theorem 1.2 of [7]

Corollary 2

Let $M^{n}$ be a complete maximal spacelike hypersurface immersed in $\mathbb {H}^{n+1}_{1}$. If there exists a vector $a\in \mathbb {R}^{n+2}_{2}$ such that $a^N$ does not vanish on $M^{n}$ and $|a^{\top }|\in \mathcal {L}^{1}(M)$, then $M^{n}$ is a totally geodesic hyperbolic space.

4 Submanifolds with parallel mean curvature vector in $\mathbb {H}^{n+p}_{p}$

In this section, we study the rigidity of a complete spacelike submanifold $M^n$ of $\mathbb {H}^{n+p}_{p}$ with nonzero parallel mean curvature vector H. For this, fixed a nonzero vector $a\in \mathbb {R}^{n+p+1}_{p+1}$, we observe that Eq. (3.4) gives us

$$\begin{aligned} \mathrm{div}\left( \langle a,\mathbf{H}\rangle a^{\top }\right)= & {} \dfrac{1}{n}\mathrm{tr}(A_{a^{N}})^{2}+\langle a,x\rangle \mathrm{tr}(A_{a^{N}})\nonumber \\&-\langle \alpha (a^{\top },a^{\top }),\mathbf{H}\rangle +\langle a,\nabla ^{\perp }_{a^{\top }}{} \mathbf{H}\rangle . \end{aligned}$$

(4.1)

Thus, from (3.10) jointly with (4.1) we obtain

$$\begin{aligned} \mathrm{div}\left[ (A_{a^{N}}-\langle a,\mathbf{H}\rangle I)a^{\top }\right]= & {} (n-1)\langle \nabla _{a^{\top }}^{\perp }{} \mathbf{H},a^{N}\rangle +\mathrm{tr}(A_{a^{N}}^{2})\nonumber \\&\!\!-\dfrac{1}{n}\mathrm{tr}(A_{a^{N}})^{2}+T(a^{\top },a^{\top }), \end{aligned}$$

(4.2)

where I denotes the identity operator in the algebra of smooth vector fields on $M^n$ and, following the terminology established in [4], T stands for a covariant tensor on $M^n$ which is given by

$$\begin{aligned} T(X,X)=\mathrm{Ric}(X,X)+(n-1)\big |X\big |^{2}-(n-1)\langle \alpha (X,X),\mathbf{H}\rangle . \end{aligned}$$

(4.3)

According to [4, 11], a spacelike submanifold $M^{n}$ of $\mathbb {H}^{n+p}_{p}$ with nonzero mean curvature vector $\mathbf{H}$ is said pseudo-umbilical if $\mathbf{H}$ is an umbilical direction. From (2.6) we see that a totally umbilical spacelike submanifold is always pseudo-umbilical. Conversely, we get

Proposition 1

Let $M^{n}$ be a complete pseudo-umbilical spacelike submanifold with nonzero parallel mean curvature vector $\mathbf{H}$ in $\mathbb {H}^{n+p}_{p}$. If there exist p vectors $a_1,\ldots ,a_p\in \mathbb {R}^{n+p+1}_{p+1}$ such that $a_1^N,\ldots ,a_p^N$ are linearly independent, with $\langle a_i,\mathbf{H}\rangle$ and $A_{a_i^{N}}$ bounded on $M^{n}$ and $|a_i^{\top }|\in \mathcal {L}^{1}(M)$ for each $1\le i\le p$, then $M^{n}$ is totally umbilical.

Proof

Let us consider $a=a_i$ for some $i\in \{1,\dots ,p\}$. We have that

$$\begin{aligned} \big |(A_{a^{N}}-\langle a,\mathbf{H}\rangle I)a^{\top }\big |\le \left( |A_{a^{N}}|+|\langle a,\mathbf{H}\rangle |\right) |a^{\top }|\le C_{2}|a^{\top }|\in \mathcal {L}^{1}(M). \end{aligned}$$

(4.4)

Since we are assuming that $M^{n}$ is pseudo-umbilical of $\mathbb {H}^{n+p}_{p}$, Lemma 4.1 of [4] assures that

$$\begin{aligned} \mathrm{Ric}(X,X)\ge -(n-1)|X|^{2}+(n-1)\langle \alpha (X,X),\mathbf{H}\rangle , \end{aligned}$$

(4.5)

for all $X\in \mathfrak {X}(M)$. Thus, from (4.3) and  (4.5) we get that $T(a^{\top },a^{\top })\ge 0$. Moreover, we observe that the function $u=\mathrm{tr}(A_{a^{N}}^{2})-\dfrac{1}{n}\mathrm{tr}(A_{a^{N}})^{2}$ is always nonnegative with $u=0$ if, and only if, $a^{N}$ is a umbilical direction. From (4.2), we obtain

$$\begin{aligned} \mathrm{div}\left[ (A_{a^{N}}-\langle a,\mathbf{H}\rangle I)a^{\top }\right] \ge 0. \end{aligned}$$

(4.6)

Thus, from (4.4) and (4.6), Lemma 1 assure us

$$\begin{aligned} \mathrm{tr}(A_{a^{N}}^{2})-\dfrac{1}{n}\mathrm{tr}(A_{a^{N}})^{2}+T(a^{\top },a^{\top })=0. \end{aligned}$$

Then, $\mathrm{tr}(A_{a^{N}}^{2})-\dfrac{1}{n}\mathrm{tr}(A_{a^{N}})^{2}=0$ and, hence, $a^{N}$ is a umbilical direction of $M^n$. Therefore, since we are supposing the existence of such vectors $a_1,\ldots ,a_p\in \mathbb {R}^{n+p+1}_{p+1}$ whose normal projections $a_1^N,\ldots ,a_p^N$ with respect to $M^n$ are linearly independent, we conclude that (2.6) holds, that is, $M^{n}$ must be totally umbilical. $\square$

Proceeding, we establish sufficient conditions to guarantee that a spacelike submanifold immersed in $\mathbb {H}^{n+p}_{p}$ with nonzero parallel mean curvature vector must be pseudo-umbilical.

Theorem 2

Let $M^{n}$ be a complete spacelike submanifold immersed in $\mathbb {H}^{n+p}_{p}$ with nonzero parallel mean curvature vector $\mathbf{H}$ and bounded normalized scalar curvature R. If there exists a nonzero vector $a\in \mathbb {R}^{n+p+1}_{p+1}$ such that $a^{N}$ is timelike, i.e., $\langle a^{N},a^{N}\rangle <0$, collinear to $\mathbf{H}$ and $|a^{\top }|\in \mathcal {L}^{1}(M)$, then $M^{n}$ is pseudo-umbilical.

Proof

Initially, taking a local orthonormal frame $\{e_{1},\ldots ,e_{n}\}$ on $M^n$, from (3.7) we get that the squared norm of second form fundamental $\alpha$ of $M^{n}$ satisfies

$$\begin{aligned} |\alpha |^{2}=\sum _{i,j}|\alpha (e_{i},e_{j})|^{2}=n^{2}\langle \mathbf{H},\mathbf{H}\rangle +n(n-1)(R+1). \end{aligned}$$

(4.7)

Now, let us consider a nonzero vector $a\in \mathbb {R}^{n+p+1}_{p+1}$ such that $a^{N}$ is timelike, collinear to $\mathbf{H}$ and with $|a^{\top }|\in \mathcal {L}^{1}(M)$. Since $M^{n}$ has bounded normalized scalar curvature and nonzero parallel mean curvature vector $\mathbf{H}$, from (4.7) we conclude that $|\alpha |^{2}$ is bounded on $M^{n}$. So, taking $\xi =\mathbf{H}$ in (3.6) we get

$$\begin{aligned} \mathrm{div}(A_{\mathbf{H}}a^{\top })=\mathrm{tr}(A_{a^{N}}\circ A_{\mathbf{H}})+\langle a,x\rangle \mathrm{tr}(A_{\mathbf{H}}), \end{aligned}$$

(4.8)

where $A_{\mathbf{H}}$ denotes the Weingarten operator associated to $\mathbf{H}$.

On the other hand, from (3.4) we have

$$\begin{aligned} \langle a,x\rangle =\dfrac{1}{n}\mathrm{div}(a^{\top })-\langle a,\mathbf{H}\rangle . \end{aligned}$$

(4.9)

Consequently, from (4.8) and (4.9)

$$\begin{aligned} \mathrm{div}(A_{\mathbf{H}}a^{\top })=\mathrm{tr}(A_{a^{N}}\circ A_{\mathbf{H}})+\mathrm{tr}(A_{\mathbf{H}})\dfrac{1}{n}\mathrm{div}(a^{\top })-\dfrac{1}{n}\mathrm{tr}({A_{a^{N}}})\mathrm{tr}(A_{\mathbf{H}}). \end{aligned}$$

(4.10)

Since

$$\begin{aligned} \mathrm{div}\left( \mathrm{tr}(A_{\mathbf{H}})a^{\top }\right) =\mathrm{tr}(A_{\mathbf{H}})\mathrm{div}(a^{\top }), \end{aligned}$$

(4.11)

from (4.10) and (4.11) we obtain

$$\begin{aligned} \mathrm{div}\,V=\mathrm{tr}(A_{a^{N}}\circ A_{\mathbf{H}})-\dfrac{1}{n}\mathrm{tr}({A_{a^{N}}})\mathrm{tr}(A_{\mathbf{H}}), \end{aligned}$$

(4.12)

where V is a tangent vector field on $M^n$ given by

$$\begin{aligned} V=\left( A_{\mathbf{H}}-\dfrac{1}{n}\mathrm{tr}(A_{\mathbf{H}})I\right) a^{\top }. \end{aligned}$$

We note that, since we are supposing $a^N$ timelike and collinear to $\mathbf{H}$, there exists on $M^n$ a smooth function $\lambda$ having strict sign such that $a^{N}=\lambda \mathbf{H}$. Thus, from (2.3) and (4.12) we get

$$\begin{aligned} \mathrm{div}\,V=\lambda \left( \mathrm{tr}(A^{2}_{\mathbf{H}})-\dfrac{1}{n}\mathrm{tr}(A_{\mathbf{H}})^{2}\right) . \end{aligned}$$

(4.13)

Consequently, from (4.13) we conclude that $\mathrm{div}\,V$ does not change sign on $M^n$. Moreover, we also have that

$$\begin{aligned} |V|\le \left( |A_{\mathbf{H}}|+|\langle \mathbf{H},\mathbf{H}\rangle |\right) |a^{\top }|\in \mathcal {L}^{1}(M). \end{aligned}$$

Hence, we can apply once more Lemma 1 to assure that $\mathrm{div}\,V=0$ on $M^n$.

Therefore, returning to (4.13) we obtain that

$$\begin{aligned} \lambda \left( \mathrm{tr}(A^{2}_{\mathbf{H}})-\dfrac{1}{n}\mathrm{tr}(A_{\mathbf{H}})^{2}\right) =0, \end{aligned}$$

which implies that $\mathbf{H}$ is an umbilical direction. $\square$

We observe that, in the case $p=1$, the notion of pseudo-umbilical coincides with that of totally umbilical. Moreover, we note that the hypothesis that $a^{N}$ is timelike amounts to the support function $f_a=\langle a,\nu \rangle$ having strict sign on the spacelike hypersurface $M^n\hookrightarrow \mathbb {H}^{n+1}_{1}$, where $\nu$ stands for the Gauss mapping of $M^n$. Consequently, taking into account the classification of the totally umbilical hypersurfaces of $\mathbb {H}^{n+1}_{1}$ (see, for instance, Example 1 of [24]) and that Theorem 1 of [21] assures us that a complete constant mean curvature spacelike hypersurface of $\mathbb {H}^{n+1}_{1}$ must have bounded second fundamental form (or, equivalently, bounded normalized scalar curvature), from Theorem 2 we obtain the following

Corollary 3

Let $M^{n}$ be a complete spacelike hypersurface immersed in $\mathbb {H}^{n+1}_{1}$ with nonzero constant mean curvature. If there exists a nonzero vector $a\in \mathbb {R}^{n+2}_{2}$ such that the support function $f_a$ has strict sign on $M^n$ and $|a^{\top }|\in \mathcal {L}^{1}(M)$, then $M^{n}$ is a totally umbilical hyperbolic space.