Nonexistence and umbilicity of spacelike submanifolds with second fundamental form locally timelike

Abstract

Under the assumption that the second fundamental form is locally timelike, we establish new nonexistence and umbilicity results concerning n-dimensional spacelike submanifolds immersed with parallel mean curvature vector in the $(n+p)$-dimensional de Sitter space $\mathbb {S}^{n+p}_q$ of index q, such that $1\le q\le p$. Our approach is based on a Simon’s type inequality involving the norm of the total umbilicity tensor, obtained by Mariano in [17], jointly with suitable maximum principles due to Alías, Caminha and do Nascimento [6, 7] for complete noncompact Riemannian manifolds and a weak version of Omori–Yau’s maximum principle for stochastically complete Riemanian manifolds proved by Pigola, Rigoli and Setti [20, 21].

1 Introduction

The study of spacelike submanifolds immersed in a semi-Riemannian manifold is a classical thematic, but it still has gotten considerable interest motivated by the nice geometric properties of these submanifolds. When the ambient space is the de Sitter space $\mathbb S^{n+1}_1$, Goddard [14] conjectured that every complete spacelike hypersurface with constant mean curvature H in $\mathbb S^{n+1}_1$ should be totally umbilical. Although the conjecture turned out to be false in its original statement, it motivated a great deal of works of several authors trying to find a positive answer to the conjecture under appropriate additional hypotheses. For instance, in 1987, Ramanathan [22] proved that Goddard’s conjecture is true for $\mathbb {S}_1^{3}$ and $0\le H\le 1$. However, for $H>1$, he showed that the conjecture is false, as it can be seen from an example due to Dajczer and Nomizu in [12]. Simultaneously and independently, Akutagawa [2] also proved that Goddard’s conjecture is true when either $n=2$ and $H^2\le 1$ or $n\ge 3$ and $H^2<\frac{4(n-1)}{n^2}$. Moreover, he also constructed complete rotational spacelike surfaces in $\mathbb {S}_1^{3}$ with constant mean curvature H satisfying $H>1$ and which are not totally umbilical. In [18], Montiel proved that Goddard’s conjecture is true provided that $M^n$ is compact (without boundary). Furthermore, he exhibited examples of complete spacelike hypersurfaces in $\mathbb {S}_1^{n+1}$ with constant mean curvature H satisfying $H^2\ge \frac{4(n-1)}{n^2}$, the so-called hyperbolic cylinders, which are isometric to a Riemannian product of the type $\mathbb {H}^{1}(r)\times \mathbb {S}^{n-1}(\sqrt{1+r^2})$, for some $r>0$. In [19], Montiel characterized these hyperbolic cylinders as the only complete noncompact spacelike hypersurfaces in $\mathbb S_1^{n+1}$ with constant mean curvature $H^2=4(n-1)/n^2$ and having at least two ends.

Regarding higher codimension, Cheng [11] extended Akutagawa’s result for n-dimensional complete spacelike submanifolds with parallel mean curvature vector field in de Sitter space $\mathbb {S}_p^{n+p}$ of index p. Afterwards, Aiyama [1] studied compact spacelike submanifolds $M^n$ in $\mathbb {S}_p^{n+p}$ with parallel mean curvature vector field, and proved that if the normal connection of $M^n$ is flat, then $M^n$ is totally umbilical. In the same work [1], she proved that compact spacelike submanifolds in $\mathbb {S}_p^{n+p}$ with parallel mean curvature vector field and nonnegative sectional curvatures must be also totally umbilical. Meanwhile, Alías and Romero [4] introduced a new method to study n-dimensional compact spacelike submanifolds in de Sitter space $\mathbb {S}_q^{n+p}$ with index $1\le q\le p$, using certain integral formulas which have a very clear geometric meaning. In particular, they got a uniqueness result for compact spacelike surfaces in $\mathbb {S}_q^{2+p}$ with parallel mean curvature vector field. Next, Li [15, 16] showed that Montiel’s result [19] still holds for higher codimensional spacelike submanifolds in $\mathbb {S}_p^{n+p}$. Later on, Camargo, Chaves, and Sousa [9] studied complete spacelike submanifolds with parallel normalized mean curvature vector field and constant scalar curvature immersed in a semi-Riemannian space form. In particular, they obtained characterization results concerning total umbilical spacelike submanifolds and hyperbolic cylinders of $\mathbb {S}_p^{n+p}$, under certain constraints on both the squared norm of the second fundamental form and on the values of the mean curvature.

In [17], Mariano established characterization results related to complete spacelike submanifolds $M^n$ with parallel mean curvature and having locally timelike second fundamental form in a semi-Riemannian space form $N^{n+p}_q(c)$ of index $1\le q\le p$, via a Simon’s type inequality for the total umbilicity tensor $\Phi$. Later, Yang and Li [23] applied the Omori–Yau’s maximum principle to get other characterization results concerning complete spacelike submanifolds with parallel mean curvature vector in $\mathbb S_q^{n+p}$. In [5], the first author jointly with Alías and dos Santos also investigated complete spacelike submanifolds $M^n$ immersed in $\mathbb S_p^{n+p}$ with parallel normalized mean curvature vector field and constant scalar curvature R. Imposing a suitable restriction on the values of R, they applied a maximum principle for the Cheng–Yau operator, which enabled them to show that either such a spacelike submanifold must be totally umbilical or it holds a sharp estimate for the norm of its total umbilicity tensor, with equality if and only the submanifold is isometric to a hyperbolic cylinder of the ambient space. In particular, when $n=2$, they also provided a nice characterization of the totally umbilical spacelike surfaces of $\mathbb {S}^{2+p}_p$ with codimension $p\ge 2$. Next, Chen, Liu, and Shu [10] obtained some integral inequalities of Simons type and rigidity theorems concerning compact spacelike submanifolds of $\mathbb S_q^{n+p}$. Afterwards, the first and second authors jointly with dos Santos [13] extended a technique developed by Alías and Meléndez [3] to prove a sharp integral inequality involving the norm of the total umbilicity tensor $\Phi$ of a compact spacelike submanifold with constant scalar curvature immersed with parallel normalized mean curvature vector field in $\mathbb S_p^{n+p}$ and they used it to characterize totally umbilical round spheres $\mathbb S^{n}(r)$ of $\mathbb S_1^{n+1}\hookrightarrow \mathbb S_p^{n+p}$.

Motivated by these works, the purpose of the present paper is to investigate the nonexistence and umbilicity of n-dimensional ($n\ge 3$) spacelike submanifolds immersed with parallel mean curvature vector in the $(n+p)$-dimensional de Sitter space $\mathbb {S}^{n+p}_q$ of index $1\le q\le p$. First, we show that there does not exist an n-dimensional complete spacelike submanifold $M^{n}$ immersed with parallel mean curvature vector, whose the second fundamental form is locally timelike in $\mathbb {S}^{n+p}_q$ and the mean curvature H satisfies $\dfrac{4(n-1)}{Q(p)}<H^{2}<1$, where $Q(x)=(n-2)^2x+4(n-1)$, such that either $|\nabla \Phi |$ is bounded and $M^n$ has polynomial volume growth or $M^n$ is noncompact and $|\Phi |$ converges to zero at infinity (see Theorem 4.1). Afterwards, we show that a complete noncompact submanifold of $\mathbb {S}_p^{n+p}$ with $H^{2}=\dfrac{4(n-1)}{Q(p)}$, such that $|\Phi |$ converges to zero at infinity, must be a totally umbilical submanifold (see Theorem 4.2). Next, we suppose that the spacelike submanifold $M^n$ is stochastically complete to show that if $H^2<1$, then either $M^n$ is totally umbilical or $\sup _M|\Phi |\ge \vartheta ^{+}_H$, where $\vartheta ^{+}_H$ is the positive root of the polynomial $P_{H,q}(x)$ defined in (3.1) (see Theorem 4.3). Finally, we prove that the only n-dimensional stochastically complete spacelike submanifold immersed in $\mathbb {S}^{n+p}_q$, which are maximal and having locally timelike second fundamental form, are the totally geodesic ones (see Theorem 4.5). Our approach is based on a Simon’s type inequality involving the norm of the total umbilicity tensor, obtained by Mariano in [17], jointly with suitable maximum principles due to Alías, Caminha and do Nascimento [6, 7] for complete noncompact Riemannian manifolds and a weak version of Omori–Yau’s maximum principle for stochastically complete Riemanian manifolds proved by Pigola, Rigoli and Setti [20, 21] (see Sect. 3). Before, in Sect. 2, we recall some basic facts related to spacelike submanifolds of the de Sitter space $\mathbb {S}^{n+p}_q$.

2 Spacelike submanifolds in the de Sitter space

Let $M^n$ be an n-dimensional (connected) spacelike submanifold isometrically immersed into the de Sitter space $\mathbb {S}_q^{n+p}$ of index $1\le q\le p$, meaning that the induced metric on $M^n$ via immersion is a Riemannian metric. In this setting, we choose a local field of pseudo-Riemannian orthonormal frame $\{e_{1},\ldots ,e_{n+p}\}$ in $\mathbb S_q^{n+p}$, such that, at each point of $M^{n}$, $e_{1},\ldots ,e_{n}$ are tangent to $M^{n}$ and $e_{n+1},\ldots ,e_{n+p}$ are normal to $M^{n}$. We use the following convention of indices:

$$\begin{aligned} 1\le A,B,C,\ldots \le n+p,\quad 1\le i,j,k,\ldots \le n\quad \text{ and }\quad n+1\le \alpha ,\beta ,\gamma ,\ldots \le n+p. \end{aligned}$$

Let $\{\omega _{1},\ldots ,\omega _{n+p}\}$ be the dual frame field of $\{e_{1},\ldots ,e_{n+p}\}$, so that the semi-Riemannian metric of $\mathbb {S}_{q}^{n+p}$ is given by $ds^{2}=\sum _{A}\epsilon _{A}\,\omega _{A}^{2}$, where $\epsilon _{A}=1$, if $1\le A\le n+p-q$, and $\epsilon _{A}=-1$, if $n+p-q+1\le A\le n+p$. Denoting by $\{\omega _{AB}\}$ the connection 1-forms of $\mathbb {S}_{q}^{n+p}$, we have that the structure equations of $\mathbb {S}_{q}^{n+p}$ are given by

$$\begin{aligned} d\omega _{A}=-\sum _{B}\epsilon _{B}\,\omega _{AB}\wedge \omega _{B},\quad \epsilon _{B}\omega _{AB}+\epsilon _{A}\omega _{BA}=0,\quad \text{ for } \text{ all } \quad 1\le A,B\le n+p, \end{aligned}$$

(2.1)

and

$$\begin{aligned} d\omega _{AB}=-\sum _{C}\epsilon _{C}\,\omega _{AC}\wedge \omega _{CB}-\frac{1}{2}\sum _{C,D}\epsilon _{C}\epsilon _{D}K_{ABCD}\,\omega _C\wedge \omega _D, \end{aligned}$$

(2.2)

where $K_{ABCD}=\epsilon _{A}\epsilon _{B}(\delta _{AD}\delta _{BC}-\delta _{AC}\delta _{BD})$.

Restricting those forms to $M^{n}$, we note that $\omega _{\alpha }=0$ for $n+1\le \alpha \le n+p$ and, hence, the Riemannian metric of $M^{n}$ is written as $ds^{2}=\sum _{i}\omega _{i}^{2}$. Since $\displaystyle \sum _{i}\omega _{\alpha i}\wedge \omega _i=d\omega _{\alpha }=0$, from Cartan’s Lemma, we can write

$$\begin{aligned} \omega _{\alpha i}=\sum _{j}h_{ij}^{\alpha }\omega _{j},\quad h_{ij}^{\alpha }=h_{ji}^{\alpha }. \end{aligned}$$

(2.3)

This gives the second fundamental form of $M^{n}$, $\displaystyle A=\sum _{\alpha ,i,j}\epsilon _{\alpha }h_{ij}^{\alpha }\omega _{i}\omega _{j}e_{\alpha }$ and the square of its length $|A|^2=\left| \sum _{\alpha }\epsilon _{\alpha }\sum _{i,j}(h_{ij}^{\alpha })^{2}\right|$. Moreover, we define the mean curvature vector and the mean curvature function on $M^n$, respectively, by

$$\begin{aligned} h:=\frac{1}{n}\sum _{\alpha }\left( \sum _{i}h_{ii}^{\alpha }\right) e_{\alpha } \ \quad \ \text{ and } \quad \ H:= |h|=\frac{1}{n}\sqrt{\sum _{\alpha }\epsilon _{\alpha }\left( \sum _{i}h_{ii}^{\alpha }\right) ^{2}}. \end{aligned}$$

In particular, $M^n$ is called maximal when its mean curvature vector h vanishes identically.

From (2.1) and (2.2), we get the structure equations of $M^{n}$

$$\begin{aligned} d\omega _{i}=-\sum _{j}\omega _{ij}\wedge \omega _{j}, \quad \omega _{ij}+\omega _{ji}=0, \ \ \text{ and } \ \ d\omega _{ij}=-\sum _{k}\omega _{ik}\wedge \omega _{kj}-\frac{1}{2}\sum _{k,l}R_{ijkl}\omega _{k}\wedge \omega _{l}, \end{aligned}$$

(2.4)

where $R_{ijkl}$ are the components of the curvature tensor of $M^{n}$. Therefore, from (2.4), we obtain the Gauss equation

$$\begin{aligned} R_{ijkl}=(\delta _{il}\delta _{jk}-\delta _{ik}\delta _{jl})+\sum _{\alpha }\epsilon _{\alpha }(h_{il}^{\alpha }h_{jk}^{\alpha }-h_{ik}^{\alpha }h_{jl}^{\alpha }). \end{aligned}$$

The components of the Ricci curvature $R_{ij}$ and the normalized scalar curvature R of $M^{n}$ are given, respectively, by

$$\begin{aligned} R_{ij}=(n-1)\delta _{ij}+\sum _{\alpha }\epsilon _{\alpha }\left\{ \left( \sum _{k}h_{kk}^{\alpha }\right) h_{ij}^{\alpha }-\sum _{\alpha ,k}h_{ik}^{\alpha }h_{kj}^{\alpha }\right\} \end{aligned}$$

and

$$\begin{aligned} R=n(n-1)+\sum _{\alpha }\epsilon _{\alpha }\left( \sum _{i}h_{ii}^{\alpha }\right) ^2-\sum _{\alpha }\sum _{i,j}\epsilon _{\alpha }(h_{ij}^{\alpha })^2. \end{aligned}$$

We also have the structure equations of the normal bundle of $M^{n}$ given by

$$\begin{aligned} d\omega _{\alpha }=-\sum _{\beta }\omega _{\alpha \beta }\wedge \omega _{\beta }, \quad \omega _{\alpha \beta }+\omega _{\beta \alpha }=0 \ \text{ and } \ d\omega _{\alpha \beta }=-\sum _{\gamma }\omega _{\alpha \gamma }\wedge \omega _{\gamma \beta }-\frac{1}{2}\sum _{k,l}R_{\alpha \beta kl}\omega _{k}\wedge \omega _{l}, \end{aligned}$$

where the components $R_{\alpha \beta jk}$ satisfy the Ricci equation

$$\begin{aligned} R_{\alpha \beta ij}=\sum _{l}\left( h_{il}^{\alpha }h_{lj}^{\beta }-h_{jl}^{\alpha }h_{li}^{\beta }\right) . \end{aligned}$$

Moreover, from (2.3), we obtain Codazzi equation

$$\begin{aligned} h_{ijk}^{\alpha }=h_{ikj}^{\alpha }=h_{kij}^{\alpha }, \end{aligned}$$

(2.5)

where $h_{ijk}^{\alpha }$ are the components of the covariant derivative $\nabla A$, which satisfy

$$\begin{aligned} \sum _{k}h_{ijk}^{\alpha }\omega _{k}=dh_{ij}^{\alpha }-\sum _{k}h_{ik}^{\alpha }\omega _{kj}-\sum _{k}h_{jk}^{\alpha }\omega _{ki}+\sum _{\beta }\epsilon _{\beta }\epsilon _{\alpha }h_{ij}^{\beta }\omega _{\beta \alpha }. \end{aligned}$$

(2.6)

Taking the exterior derivative in (2.6), we obtain the following Ricci formula for the second fundamental form

$$\begin{aligned} h_{ijkl}^{\alpha }-h_{ijlk}^{\alpha }=-\sum _{m}h_{mj}^{\alpha }R_{mikl}-\sum _{m}h_{im}^{\alpha }R_{mjkl}+\sum _{k,\beta }\epsilon _{\beta }\epsilon _{\alpha }h_{ik}^{\beta }R_{\alpha \beta jk}. \end{aligned}$$

(2.7)

The Laplacian $\Delta h_{ij}^{\alpha }$ of the components $h_{ij}^{\alpha }$ of second fundamental form is defined by

$$\begin{aligned} \displaystyle \Delta h_{ij}^{\alpha }:=\sum _{k}h_{ijkk}^{\alpha }. \end{aligned}$$

Therefore, from Eqs. (2.5) and (2.7), we get the following formula:

$$\begin{aligned} \Delta h_{ij}^{\alpha }=\sum _{k}h_{kkij}^{\alpha }-\sum _{k,l}h_{kl}^{\alpha }R_{lijk}-\sum _{k,l}h_{li}^{\alpha }R_{lkjk}+\sum _{k,\beta }\epsilon _{\beta }\epsilon _{\alpha }h_{ik}^{\beta }R_{\alpha \beta jk}. \end{aligned}$$

We will assume that the mean curvature vector h is parallel as a section of the normal bundle of $M^n$, which means that $\nabla ^{\perp }h=0$, where $\nabla ^{\perp }$ is the normal connection of $M^n$. Considering $H>0$, we can assume that the orthonormal frame $\{e_{1},\ldots ,e_{n+p}\}$ in $\mathbb S_q^{n+p}$ is such that $e_{n+p-q+1}=\frac{h}{H}$. Consequently, we get

$$\begin{aligned} H^{n+1}:=\dfrac{1}{n}\textrm{tr}(h^{n+p-q+1})=H\quad \text{ and }\quad H^{\alpha }:=\dfrac{1}{n}\textrm{tr}(h^{\alpha })=0,\,\alpha \ne n+p-q+1, \end{aligned}$$

where $h^{\alpha }$ denotes the matrix $(h^{\alpha }_{ij})$.

Furthermore, we will also consider the total umbilicity tensor

$$\begin{aligned} \Phi =\sum _{i,j,\alpha \ge n+p-q+1}\Phi ^\alpha _{ij}\omega _i\otimes \omega _je_\alpha , \end{aligned}$$

(2.8)

where $\Phi ^\alpha _{ij}=h^\alpha _{ij}-H^\alpha \delta _{ij}$. We have that

$$\begin{aligned} \Phi ^{n+p-q+1}_{ij}=h^{n+p-q+1}_{ij}-H\delta _{ij} \quad \text {and} \quad \Phi ^\alpha _{ij}=h^\alpha _{ij}, \end{aligned}$$

for $\alpha \ne n+p-q+1$. Since $|\Phi |^2=\sum _{\alpha ,i,j}(\Phi ^\alpha _{ij})^2$ is the square of the length of $\Phi$, it is not difficult to verify that $\Phi$ is traceless with

$$\begin{aligned} |\Phi |^2=|A|^2-nH^2. \end{aligned}$$

Besides, we observe that $|\Phi |$ vanishes identically on $M^n$ if and only if $M^n$ is a totally umbilical submanifold of $\mathbb {S}_q^{n+p}$.

3 Auxiliaries results

This section is devoted to quote some auxiliary lemmas which will be used to prove our main results in the next section. The first one is the following result given in [17, Lemma 3.1].

Lemma 3.1

Let $Q(x)=(n-2)^2x+4(n-1)$ and $P_{H,q}(x)$ be the polynomial defined by

$$\begin{aligned} P_{H,q}(x)=\frac{x^2}{q}-\frac{n(n-2)}{\sqrt{n(n-1)}}Hx+n(1-H^2). \end{aligned}$$

(3.1)

Then

• 1. If $H^2<\frac{4(n-1)}{Q(q)}$, then $P_{H,q}(x)>0$ for any $x\in \mathbb {R}$.

• 2. If $H^2=\frac{4(n-1)}{Q(q)}$, then the (double) root of $P_{H,q}(x)$ is

  $$\begin{aligned} \vartheta ^{\pm }_H=\frac{n(n-2)q}{\sqrt{n}}\sqrt{\frac{1}{Q(q)}}, \end{aligned}$$

  and so $P_{H,q}(x)=\left( x-\frac{n(n-2)q}{\sqrt{n}}\sqrt{\frac{1}{Q(q)}}\right) ^2\ge 0$.

• 3. If $H^2>\frac{4(n-1)}{Q(q)}$, then $P_{H,q}(x)$ has two real roots $\vartheta ^{+}_H$ and $\vartheta ^{-}_H$ given by

  $$\begin{aligned} \vartheta ^{\pm }_H=p\sqrt{\frac{n}{4(n-1)}}\left\{ (n-2)H\pm \sqrt{\frac{Q(p)H^2-4(n-1)}{q}}\right\} . \end{aligned}$$

  We have that $\vartheta ^{+}_H$ is always positive, $\vartheta ^{-}_H<0$ if and only if $H^2=1$ and $\vartheta ^{-}_H>0$ if and only if $\frac{4(n-1)}{Q(q)}\le H^2<1$.

Moreover, when $H^2=\frac{4(n-1)}{Q(p)}$, we have that $P_{H,q}(x)$ is strictly decreasing for all $x\le \vartheta ^{\pm }_H$ and, when $H^2>\dfrac{4(n-1)}{Q(p)}$, $P_{H,q}(x)$ is strictly decreasing for all $x\le \vartheta ^{-}_H$.

We will also need the following Simon’s type inequality involving the norm of the total umbilicity tensor, which is deduced in [17, Lemma 3.2]. At this point, we draw attention that in the proof of this inequality, it is not necessary assuming the hypothesis of the spacelike submanifold be complete. This fact will be used to establish our last result in the next section.

Lemma 3.2

Let $M^n$ be a spacelike submanifold immersed with parallel mean curvature vector in $\mathbb {S}^{n+p}_q(c)$, $1\le q\le p$, and such that its second fundamental form is locally timelike. Then, the following inequality holds:

$$\begin{aligned} \frac{1}{2}\Delta |\Phi |^2\ge |\Phi |^2P_{H,q}(|\Phi |), \end{aligned}$$

where $P_{H,q}(x)$ is the polynomial defined in (3.1).

In what follows, we present some concepts and maximum principles which will be used to prove our main results in the next section. First, let $(M^n,\langle \,,\rangle )$ be a connected, oriented, complete Riemannian manifold. We denote by B(p, t) the geodesic ball centered at p with radius t. Given a polynomial function $\sigma :(0,+\infty )\rightarrow (0,+\infty )$, we say that $M^n$ has polynomial volume growth like $\sigma (t)$ if there exists $p\in M^n$, such that

$$\begin{aligned} \textrm{vol}(B(p,t))=\mathcal {O}(\sigma (t)), \end{aligned}$$

as $t\rightarrow +\infty$, where $\textrm{vol}$ denotes the standard Riemannian volume. As it was observed in [7, Section 2], if $p,q\in M^n$ are at distance d from each other, we can verify that

$$\begin{aligned} \dfrac{\textrm{vol}(B(p,t))}{\sigma (t)}\ge \dfrac{\textrm{vol}(B(q,t-d))}{\sigma (t-d)}\cdot \dfrac{\sigma (t-d)}{\sigma (t)}. \end{aligned}$$

Consequently, the choice of p in the notion of volume growth is immaterial. For this reason, we will just say that $M^n$ has polynomial volume growth.

Keeping in mind this previous digression and denoting by $\textrm{div}X$ the divergence of a smooth vector field $X\in \mathfrak {X}(M)$ in the metric $\langle \,,\rangle$, we quote the following key lemma which corresponds to a particular case of [7, Theorem 2.1].

Lemma 3.3

Let $(M^n,\langle \,,\rangle )$ be a connected, oriented, complete noncompact Riemannian manifold and let $X\in \mathfrak {X}(M)$ be a bounded smooth vector field on $M^n$. Assume that $f\in C^{\infty }(M)$ is a smooth function on $M^n$, such that $\langle \nabla f,X\rangle \ge 0$ and $\textrm{div}X\ge \alpha f$, for some positive constant $\alpha$. If $M^n$ has polynomial volume growth, then $f\le 0$ on $M^n$.

The next auxiliary lemma is due to Barros et al. (see [8, Lemma 1]).

Lemma 3.4

Let $M^n$ be a Riemannian manifold isometrically immersed into a Riemannian manifold $N^{n+p}$. Consider $\Psi =\displaystyle {\sum _{\alpha ,i,j}}\Psi _{ij}^{\alpha }\omega _{i}\otimes \omega _{j}\otimes e_{\alpha }$ a traceless symmetric tensor satisfying Codazzi equation. Then, the following inequality holds:

$$\begin{aligned} |\nabla |\Psi |^2|^2\le \dfrac{4n}{n+2}|\Psi |^2|\nabla \Psi |^2, \end{aligned}$$

where $|\Psi |^2=\displaystyle \sum _{\alpha ,i,j}(\Psi _{ij}^\alpha )^2$ and $|\nabla \Psi |^2=\displaystyle \sum _{\alpha ,i,j,k}(\Psi _{ijk}^\alpha )^2$. In particular, the conclusion holds for the tensor $\Phi$ defined in (2.8).

Given a connected, complete noncompact Riemannian manifold $M^n$, we let $d(\cdot ,o):M^n\rightarrow [0,+\infty )$ stand for its Riemannian distance, measured from a fixed point $o\in M^n$. According to [6, Section 2], if $f\in C^0(M^n)$ satisfies

$$\begin{aligned} \lim _{d(x,o)\rightarrow +\infty }f(x)=0, \end{aligned}$$

we say that fconverges to zero at infinity.

In this setting, we have the following maximum principle which corresponds to [6, Theorem 2.2(a)].

Lemma 3.5

Let $(M^n,\langle \,,\rangle )$ be a connected, oriented, complete noncompact Riemannian manifold and let $X\in \mathfrak {X}(M^n)$ be a smooth vector field on $M^n$. Assume that there exists a nonnegative, non-identically vanishing function $f\in C^{\infty }(M)$ which converges to zero at infinity, such that $\langle \nabla f,X\rangle \ge 0$. If $\textrm{div}X\ge 0$ on $M^n$, then $\langle \nabla f,X\rangle \equiv 0$ on $M^n$.

We recall that a (non necessarily complete) Riemannian manifold $M^n$ is said to be stochastically complete if, for some (and, hence, for any) $(x, t)\in M^n\times (0,+\infty )$, the heat kernel p(x, y, t) of the Laplace–Beltrami operator $\Delta$ satisfies the conservation property

$$\begin{aligned} \int _{M}p(x,y,t)d\mu (y)=1. \end{aligned}$$

Pigola, Rigoli and Setti showed that stochastic completeness turns out to be equivalent to the validity of a weak version of the Omori–Yau’s maximum principle (see [20, Theorem 1.1] and [21, Theorem 3.1]), as is expressed below.

Lemma 3.6

A Riemannian manifold $M^n$ is stochastically complete if, and only if, for every $u\in C^2(M)$ satisfying $\sup _Mu<+\infty$, there exists a sequence of points $\{p_k\}\subset M^n$, such that

$$\begin{aligned} \lim _{k\rightarrow \infty }u(p_k)=\sup _Mu \quad \textrm{and} \quad \limsup _{k\rightarrow \infty }\Delta u(p_k)\le 0. \end{aligned}$$

4 Main results

Returning to the context of spacelike submanifolds immersed in the de Sitter space, we start this section obtaining the following nonexistence result.

Theorem 4.1

There does not exist an n-dimensional ($n\ge 3$) complete spacelike submanifold $M^{n}$ immersed with parallel mean curvature vector in the $(n+p)$-dimensional de Sitter space $\mathbb {S}^{n+p}_q$ of index $1\le q\le p$, such that the second fundamental form is locally timelike, $\dfrac{4(n-1)}{Q(p)}<H^{2}<1$, where $Q(x)=(n-2)^2x+4(n-1)$, and one of the following items is satisfied:

• (i) $|\nabla \Phi |$ is bounded, $\sup _M|\Phi |<\vartheta ^{-}_H$ and $M^n$ has polynomial volume growth;

• (ii) $M^n$ is noncompact and $|\Phi |$ converges to zero at infinity.

Proof

Let us suppose by contradiction the existence of such a submanifold $M^n$. Therefore, we take the smooth vector field $X=\nabla |\Phi |^2$ and the smooth function $f=|\Phi |^2$.

Assuming that $|\nabla \Phi |$ is bounded, $\sup _M|\Phi |<\vartheta ^{-}_H$ and $M^n$ has polynomial volume growth, let us observe that it will fulfill the required conditions to apply Lemma 3.3. Indeed, Lemma 3.4 guarantees that $\nabla |\Phi |^2$ is also bounded. Consequently, we get

$$\begin{aligned} |X|=|\nabla |\Phi |^2|\le C<+\infty , \end{aligned}$$

(4.1)

for some positive constant $C\in \mathbb R$.

Moreover, we also have

$$\begin{aligned} \langle \nabla f,X\rangle =|\nabla |\Phi |^2|^2\ge 0 \end{aligned}$$

(4.2)

is also verified.

On the other hand, since we are supposing that the mean curvature vector is parallel, we can consider the constant $\gamma :=H^2$. Therefore, taking into account that $\dfrac{4(n-1)}{Q(p)}<H^{2}<1$, we obtain

$$\begin{aligned} P_{H,q}(x)=\dfrac{x^2}{q}-\frac{n(n-2)}{\sqrt{n(n-1)}}Hx-n\left( H^{2}-1\right) \ge \dfrac{x^2}{q}-\frac{n(n-2)}{\sqrt{n(n-1)}}\sqrt{\gamma }x-n\left( \gamma -1\right) =:P_{\gamma ,q}(x). \end{aligned}$$

However, Lemma 3.1 guarantees that $\vartheta ^{-}_H>0$ if and only if $\frac{4(n-1)}{Q(q)}\le H^2<1$ and then, since $\sup _M|\Phi |<\vartheta ^{-}_H$ and $P_{H,q}(x)$ is strictly decreasing for $x\le \vartheta ^{-}_H$

$$\begin{aligned} P_{H,q}(|\Phi |)\ge P_{\gamma ,q}(|\Phi |)\ge P_{\gamma ,q}(\sup _M|\Phi |)>P_{\gamma ,q}(\vartheta ^{-}_H)=0. \end{aligned}$$

(4.3)

Therefore, from (4.3), we conclude that

$$\begin{aligned} \textrm{div}X=\textrm{div}(\nabla |\Phi |^2)=\Delta |\Phi |^2\ge P_{H,q}(|\Phi |)|\Phi |^2\ge \alpha f, \end{aligned}$$

(4.4)

where $\alpha =P_{\gamma ,q}(\sup _M|\Phi |)>0$.

Now, assuming that $M^n$ is noncompact, since (4.1), (4.2) and (4.4) were verified and $M^n$ has polynomial volume growth, we are able to apply Lemma 3.3 to obtain that $|\Phi |^2\equiv 0$ and, consequently, $M^n$ is a totally umbilical submanifold.

When $M^n$ is compact, we can integrate both sides of (4.4) and use the Divergence Theorem to get that

$$\begin{aligned} \int _MP_{H,q}(|\Phi |)|\Phi |^2\textrm{dM}\le \int _M\textrm{div}(\nabla |\Phi |^2)\textrm{dM}=0. \end{aligned}$$

Thus, we have $P_{H,q}(|\Phi |)|\Phi |^2=0$ and, since $P_{H,q}(|\Phi |)>0$ from (4.3), it must be $|\Phi |=0$ and, again, $M^n$ is a totally umbilical submanifold.

In the case that $M^n$ is noncompact and $|\Phi |$ converges to zero at infinity, since $P_{H,q}(|\Phi |)\ge 0$ for $|\Phi |\le \vartheta ^{-}_H$ from (4.3), Lemma 3.2 gives

$$\begin{aligned} \textrm{div}X=\textrm{div}(\nabla |\Phi |^2)=\Delta |\Phi |^2\ge P_{H,q}(|\Phi |)|\Phi |^2\ge 0. \end{aligned}$$

Consequently, we can apply Lemma 3.5 to get that

$$\begin{aligned} \langle \nabla f,X\rangle =|\nabla |\Phi |^2|^2=0, \end{aligned}$$

and conclude that $\nabla |\Phi |\equiv 0$. Thus, $f=|\Phi |$ is constant and, since f converges to zero at infinity, it must be identically zero and $M^n$ must be a totally umbilical submanifold of $\mathbb {S}_q^{n+p}$.

However, from the proof of item (d) of [17, Theorem 1.1], our constraint on the value of the mean curvature implies that $\vartheta ^{-}_H\le \sup _M|\Phi |\le \vartheta ^{+}_H$ with $\vartheta ^{-}_H>0$, leading us to a contradiction. $\square$

Proceeding, we obtain a characterization for totally umbilical spacelike submanifolds of $\mathbb {S}^{n+p}_q$.

Theorem 4.2

Let $M^{n}$ be an n-dimensional ($n\ge 3$) complete noncompact spacelike submanifold immersed with parallel mean curvature vector in the $(n+p)$-dimensional de Sitter space $\mathbb {S}^{n+p}_q$ of index $1\le q\le p$, such that the second fundamental form is locally timelike and $H^{2}=\dfrac{4(n-1)}{Q(q)}$, where $Q(x)=(n-2)^2x+4(n-1)$. If $|\Phi |$ converges to zero at infinity, then $M^n$ is a totally umbilical submanifold of $\mathbb {S}^{n+p}_q$.

Proof

Let us consider once more the smooth vector field $X=\nabla |\Phi |^2$ and the smooth function $f=|\Phi |^2$. Therefore

$$\begin{aligned} \langle \nabla f,X\rangle =|\nabla |\Phi |^2|^2\ge 0. \end{aligned}$$

Let us suppose that $M^n$ is not a umbilical submanifold. Therefore, we have that f is a non-identically vanishing function which converges to zero at infinity. Moreover, since $H^{2}=\dfrac{4(n-1)}{Q(q)}$, we have from Lemma 3.1 that $P_{H,q}\ge 0$. Hence, we can apply Lemma 3.5 to get that

$$\begin{aligned} \langle \nabla f,X\rangle =|\nabla |\Phi |^2|^2=0, \end{aligned}$$

which implies that $\nabla |\Phi |\equiv 0$. Thus, $f=|\Phi |$ is constant and, since f converges to zero at infinity, it must be identically zero, leading us to a contradiction. Therefore, $M^n$ must be a totally umbilical submanifold of $\mathbb {S}_q^{n+p}$. $\square$

Considering stochastically complete spacelike submanifolds with parallel mean curvature vector, we obtain the following result.

Theorem 4.3

Let $M^{n}$ be an n-dimensional ($n\ge 3$) stochastically complete spacelike submanifold immersed with parallel mean curvature vector in the $(n+p)$-dimensional de Sitter space $\mathbb {S}^{n+p}_q$ of index $1\le q\le p$, such that the second fundamental form is locally timelike. If $H^{2}<1$, then either $M^n$ is totally umbilical or $\sup _M|\Phi |\ge \vartheta ^{+}_H$.

Proof

From Lemma 3.1, if $H^{2}\le \dfrac{4(n-1)}{Q(p)}$, then $P_{H,q}\ge 0$. Also, $\vartheta ^{-}_H>0$ if, and only if $\dfrac{4(n-1)}{Q(p)}\le H^{2}<1.$ Hence, we have $P_{H,q}(|\Phi |)\ge 0$ for $|\Phi |\le \vartheta ^{-}_H$. Thus, from Lemma 3.2, we obtain

$$\begin{aligned} \Delta |\Phi |^2\ge P_{H,q}(|\Phi |)|\Phi |^2\ge 0, \end{aligned}$$

(4.5)

for $H^2<1$.

If $\sup _M|\Phi |^2=+\infty$, then it is immediate that $\sup _M|\Phi |\ge \vartheta ^{+}_H$. Therefore, let us suppose that $\sup _M|\Phi |^2<+\infty$. Thus, Lemma 3.6 guarantees that there exists a sequence of points $\{p_k\}\subset M^n$, such that

$$\begin{aligned} \lim _{k\rightarrow \infty }|\Phi |^2(p_k)=\sup _M|\Phi |^2 \quad \textrm{and} \quad \limsup _{k\rightarrow \infty }\Delta |\Phi |^2(p_k)\le 0. \end{aligned}$$

Consequently, taking into account the continuity of the polynomial $P_{H,q}(x)$, from (4.5), we have

$$\begin{aligned} 0\ge \frac{1}{2}\limsup _{k\rightarrow \infty }\Delta |\Phi |^2(p_k)\ge & {} \limsup _{k\rightarrow \infty }(|\Phi |^2P_{H,q}(\Phi |))(p_k)=\lim _{k\rightarrow \infty }(|\Phi |^2P_{H,q}(|\Phi |))(p_k)\\= & {} \lim _{k\rightarrow \infty }|\Phi |^2(u_k)P_{H,q}(\lim _{k\rightarrow \infty }|\Phi |(p_k))=\sup _M|\Phi |^2P_{H,q}(\sup _M|\Phi |). \end{aligned}$$

Hence, we obtain

$$\begin{aligned} \sup _M|\Phi |^2P_{H,q}(\sup _M|\Phi |)\le 0. \end{aligned}$$

(4.6)

Thus, either $\sup _M|\Phi |>0$ and then

$$\begin{aligned} P_{H,q}(\sup _M|\Phi |)\le 0, \end{aligned}$$

which implies that $\sup _M|\Phi |\ge \vartheta ^{+}_H$, or $\sup _M|\Phi |=0$, which means that $|\Phi |\equiv 0$ and $M^n$ must be totally umbilical. $\square$

We recall that a Riemannian manifold without boundary $M^n$ is said to be parabolic when the only superharmonic functions on $M^n$ bounded from below are the constant ones. Taking into account that every parabolic Riemannian manifold is stochastically complete, we obtain the following consequence of Theorem 4.3.

Corollary 4.4

Let $M^n$ be an n-dimensional ($n\ge 3$) parabolic spacelike submanifold immersed with parallel mean curvature vector in the $(n+p)$-dimensional de Sitter space $\mathbb {S}^{n+p}_q$ of index $1\le q\le p$, such that the second fundamental form is locally timelike. If $H^{2}<1$, then either $M^n$ is a totally umbilical submanifold or $\sup _{M}|\Phi |\ge \vartheta ^{+}_H$.

We close our paper extending the case $c>0$ in [17, Theorem 1.2] for the context of stochastically complete spacelike submanifolds.

Theorem 4.5

The only n-dimensional ($n\ge 3$) stochastically complete spacelike submanifold immersed in the $(n+p)$-dimensional de Sitter space $\mathbb {S}^{n+p}_q$ of index $1\le q\le p$, which are maximal and having locally timelike second fundamental form, are the totally geodesic ones.

Proof

Let be $M^n$ such a spacelike submanifold of $\mathbb {S}^{n+p}_q$. Since H is identically zero, we obtain from (3.1) that

$$\begin{aligned} P_{0,q}(\sup _M|\Phi |)=\frac{(\sup _M|\Phi |)^2}{q}+n>0. \end{aligned}$$

Hence, inequality (4.6) allows us to conclude that $\sup _M|\Phi |=0$. Therefore, $|\Phi |=0$ and $M^n$ must be totally geodesic. $\square$