Spacelike submanifolds with parallel mean curvature vector in a de Sitter space Sqn+p(c)$S^{n+p}_{q}(c)$

*Correspondence: mailjunfeng@163.com 1School of Mathematics and Statistics, Xidian University, 710126, Xi’an, P.R. China 2School of Mathematics and Information Science, Xianyang Normal University, 712000, Xianyang, P.R. China †Equal contributors Abstract Spacelike submanifolds usually appear in the study of questions related to causality in general relativity. In this paper, we study an n-dimensional spacelike submanifold in (n + p)-dimensional connected de Sitter space S q (c) of index q (1≤ q≤ p) and of constant curvature c, and we obtain some integral inequalities of Simons type and rigidity theorems.


Introduction
During the last decades, the study of spacelike submanifolds in semi-Riemannian manifolds has got increasing interest motivated by their importance in problems related to Physics, such as the theory of general relativity. Furthermore, the unique properties of spacelike submanifolds are of great significance for solving the Cauchy initial value problem of hypersurfaces and the propagation of gravity in arbitrary space-time (see, for example, [1][2][3]). Therefore, many authors have focused on the development of spacelike submanifolds in semi-Riemannian manifolds; see, for example, [4][5][6][7] and the reference therein.
Let M be a finite dimensional manifold, we assume that M can be endowed with a Riemannian metric to become a Riemannian manifold. The structure and pinching problem of some special submanifolds such as totally geodesic submanifolds, minimal submanifolds, submanifolds with parallel mean curvature vector and totally umbilical submanifolds are the research focus of submanifolds on Riemannian manifolds. The pinching problem of submanifolds is to restrict norm square of the second fundamental form, sectional curvature, Ricci curvature and scalar curvature of submanifolds, so as to obtain some special properties.
Let N n+p q (c) be an (n + p)-dimensional connected semi-Riemannian manifold with constant curvature c, and of index q, where 1 ≤ q ≤ p. It is called an indefinite space form of index q. More specifically, it may be considered, up to isometries, as de Sitter space S n+p q (c), semi-Euclidean space R n+p q , and semi-hyperbolic space H n+p q (c), if c > 0, c = 0, and © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. c < 0, respectively. A submanifold immersed in N n+p q (c) is said to be spacelike, timelike, lightlike, if the metric in M n induced by that of the ambient space N n+p q (c) is positive definite, negative definite, vanishing, respectively. As is usual, the spacelike submanifold is said to be complete if the Riemannian induced metric is a complete metric. For further details, see [8].
Let ϕ : M → N n+p q (c) be an n-dimensional spacelike submanifold in N n+p q (c). If q = p = 1, this initial step in this context is due to Goddard's conjecture in 1977 (see [9]) that complete spacelike hypersurfaces of S n+1 1 with constant mean curvature H must be totally umbilical. In order to prove this conjecture, many researchers began to study spacelike submanifolds in constant curvature space. Although the conjecture turned out to be false in its original statement, it motivated a great deal of work of several authors trying to find a positive answer to the conjecture under appropriate additional hypotheses (see [10][11][12][13][14][15]). In the case of higher codimension (i.e. q = p > 1) in N n+p q (c), several fruitful results have been achieved in recent years. Among them, Cheng [16] extended Akugatawa's result [10] to complete spacelike submanifolds with parallel mean curvature vector fields in de Sitter space S n+p p . Li [17] proved that the conclusion of Motiel [12] is still valid in spacelike submanifolds. For relevant conclusions, refer to [18][19][20][21][22][23][24][25][26].
When q = p, we note here that complete maximal spacelike submanifold M in N n+p p (c) is totally geodesic for c ≥ 0 (see [27]). Thus the class of all such submanifolds is very small. While if 0 ≤ q < p, and M is a complete minimal submanifold in sphere S m (c) (m > n), which is embedded in S m+q q (c) as a totally geodesic spacelike submanifold such that mn + q = p, we know from [28] and [29] that M is a complete maximal spacelike submanifold in S n+p q (c). This implies that the class of complete maximal spacelike submanifold in S n+p q (c) is very large. From the above discussion, it is necessary and important to study the classification of spacelike submanifold in S n+p q (c) (1 ≤ q < p). But to the best of our knowledge, the progress of this research topic is slow.
There are several authors have tried relevant topic and obtained some important properties. By calculating the divergence of certain tangent vector fields and using the divergence theorem, Alías and Romero [28] proved an integral formula for the compact spacelike n-dimensional submanifolds in a de Sitter spaces S n+p q (c) (1 ≤ q < p), and obtained a Bernstein type result for the complete maximal submanifolds in S n+p q (c) (1 ≤ q < p). Cheng and Ishikawa [29] studied compact maximal spacelike submanifold in S n+p q (c) (1 ≤ q < p), and obtained some important results in terms of the pinching conditions on scalar curvature, sectional curvature and Ricci curvature, respectively. Under the assumption that the second fundamental form of M is locally timelike, Mariano [30] obtained some results of complete spacelike submanifold with parallel mean curvature vector in S n+p q (c) (1 ≤ q < p). And Yang and Li [31] obtained some classification results for spacelike submanifold in S n+p q (c) (1 ≤ q < p), but they not only assume the mean curvature vector is parallel but also it is spacelike or timelike.
Inspired and motivated by the research work above, in this paper, only assuming the mean curvature vector is parallel, we continue to study this topic and prove some integral inequalities of Simons' type and rigidity theorems for n-dimensional spacelike submanifolds in a de Sitter space S n+p q (c) (1 ≤ q < p), which is a further generalization of the results obtained in [29].
Next, we will make a brief introduction to the main results present in this paper. We denote by ρ 2 the nonnegative function ρ 2 = S -nH 2 , where S and H are the norm square of the second fundamental form and the mean curvature vector of M, we see that ρ 2 = 0 if and only if M is a totally umbilical spacelike submanifold. We also denote by K and Q the functions which assign to each point of M the infimum of the sectional curvature and the Ricci curvature at the point, we will present the following theorems.
with parallel mean curvature vector. Then the following integral inequality holds: and is isometric to the Veronese surface.
be an n (n ≥ 2)-dimensional compact spacelike submanifold in a de Sitter space S n+p q (c) with parallel mean curvature vector. Then the following integral inequality holds: In particular, if then M is totally umbilical, or M is a spacelike submanifold with parallel second fundamental form.
be an n (n ≥ 2)-dimensional compact spacelike submanifold in a de Sitter space S n+p q (c) with parallel mean curvature vector. Then the following integral inequality holds: In particular, if then M is totally umbilical, or M is a maximal Einstein submanifold with parallel second fundamental form, and the Ricci curvature From Theorem 1.3, we also have the following corollary.
In particular, if

then M is totally geodesic, or M is a maximal Einstein submanifold with Ricci curvature
Remark 1 If H = 0, i.e. M is maximal, we see that the second part of Theorem 1.1, Theorem 1.2, and Corollary 1.4 are reduced to Theorem 1, Theorem 2 (if pq = 1) and Theorem 3 (if p = 2, q = 1) of [29], respectively. Thus, we generalize the results of [29] to spacelike submanifold with parallel mean curvature vector for any 1 ≤ q < p.

Preliminaries
In this section, we will introduce some basic facts and notations that will appear on the paper. Let Let ω 1 , . . . , ω n+p be its dual frame field, so that the semi-Riemannian metric of N n+p q (c) is given by Then the structure equations of N n+p q (c) are given by (see [29]) If we restrict these form to M, then ω α = 0 (n + 1 ≤ α ≤ n + p), and The second fundamental form II, the mean curvature vector H of M are defined by The norm square of the second fundamental form and the mean curvature of M are defined by The Gauss equations are Defining the first and the second covariant derivatives of h α ij , say h α ijk and h α ijkl by we have the Codazzi equations and the Ricci identities The Ricci equations are The Laplacian of h α ij is defined by h α ij = k h α ijkk . From (2.6), we obtain for any α (n + 1 ≤ α ≤ n + p), We need the following lemma (see [37]).

Lemma 2.1 Let A, B be symmetric n × n matrices satisfying AB = BA and tr
Then

Basic formulas
This section introduces some basic formulas which plays a crucial role in the proof of the theorems in this paper. Define the tensors Then the (p × p)-matrix (σ αβ ) is symmetric and can be assumed to be diagonalized for a suitable choice of e n+1 , . . . , e n+p . We set σ αβ =σ α δ αβ .
In general, for a matrix A = (a ij ) we denote by N(A) the square of the norm of A, that is, Since the mean curvature vector is parallel, that is, |∇ ⊥ H| 2 = i,α (H α ,i ) 2 = 0, we see that H α ,i = 0 for all i, α and H α are constant for all α, this implies that H is constant. Putting (3.6) and (3.7) into (3.5), we have

Proofs of theorems
Proof of Theorem 1.1 We first have the following: where the inequality N(Ã αÃβ -Ã βÃα ) ≥ 0 for any α, β is used.
Proof of Theorem 1.2 For a fixed α, n + 1 ≤ α ≤ n + p, we can take a local orthonormal frame field {e 1 , . . . , e n } such that h α  In particular, if K ≥ 1 n (1 -1 p-q )ρ 2 , from (4.10), we see that ρ 2 = 0 and M is totally umbilical, or K = 1 n (1-1 p-q )ρ 2 . In the latter case, we see that |∇h| = 0 and M is a spacelike submanifold with parallel second fundamental form. This completes the proof of Theorem 1.2.