Spacelike Hypersurfaces in de Sitter Space

Abstract

A closed conformal vector field in de Sitter space $S_1^{n+1}\left(\overline{c}\right)$ induces a vector field on a spacelike hypersurface M of $S_1^{n+1}\left(\overline{c}\right)$, referred to as the induced vector field on M. This article investigates the characterization of compact spacelike hypersurfaces in de Sitter space without assuming the constancy of the mean curvature. Specifically, we establish that under certain conditions, a compact spacelike hypersurface in $S_1^{n+1}\left(\overline{c}\right)$ is a sphere, that is, a totally umbilical hypersurface with constant mean curvature. We also present a different characterization of compact spacelike hypersurfaces in de Sitter space as spheres by using a lower bound on the integral of the Ricci curvature of the compact hypersurface in the direction of the induced vector field. We also consider de Sitter space as a Robertson–Walker space and provide several characterizations of spheres within its spacelike hypersurfaces.

1. Introduction

De Sitter space is a Lorentzian manifold with constant positive curvature, often represented as a hyperboloid embedded in a higher-dimensional Minkowski space. Let ${\mathbb{R}}_1^{n+2}$ be the $\left(n+2\right)$-dimensional Minkowski space, which is the real vector space ${\mathbb{R}}^{n+2}$ equipped with the Lorentzian metric $〈·,·〉$, with signature $(-,+,\cdots ,+)$, and defined for all $X,Y\in {\mathbb{R}}_1^{n+2}$ as

$$\begin{array}{c}\hfill 〈X,Y〉=-X_1{Y}_1+\sum _{i=2}^{n+2}{X}_i{Y}_i.\end{array}$$

Let $S_1^{n+1}\left(\overline{c}\right)$ be the hypersurface of ${\mathbb{R}}_1^{n+2}$ defined by

$$\begin{array}{c}\hfill S_1^{n+1}\left(\overline{c}\right)=\left\{X\in {\mathbb{R}}_1^{n+2}:〈X,X〉={\displaystyle \frac{1}{\overline{c}}}\right\},\end{array}$$

where $\overline{c}$ is a positive constant.

Then, $S_1^{n+1}\left(\overline{c}\right)$ inherits a Lorentzian metric $\overline{g}$ from ${\mathbb{R}}_1^{n+2}$ with a constant sectional curvature $\overline{c}$, and $\left(S_1^{n+1}\left(\overline{c}\right),\overline{g}\right)$ is called a de Sitter space. This is the unique geodesically complete spacetime, which is simply connected if $n\ge 2$. Thus, de Sitter space is a vacuum solution to Einstein’s equations with a positive cosmological constant. It can be depicted as a one-sheeted hyperboloid in Minkowski space and can be written in global coordinates as a warped product (refer to Section 5).

A smooth immersion $\psi :M\to S_1^{n+1}\left(\overline{c}\right)\subset {\mathbb{R}}_1^{n+2}$ of an n-dimensional connected manifold M is called a spacelike hypersurface if the metric g induced on M is a Riemannian metric. In other words, M is a submanifold of $S_1^{n+1}\left(\overline{c}\right)$ where all tangent vectors are spacelike.

Spacelike hypersurfaces in de Sitter space have been a subject of extensive research in differential geometry. The interest in studying these hypersurfaces is motivated by their implications for general relativity and theoretical physics. There are two significant types of spacelike hypersurfaces of the de Sitter space $S_1^{n+1}\left(\overline{c}\right)$: the totally geodesic hypersurfaces $S^n\left(\overline{c}\right)$, known as great spheres, and $S^n\left(c\right)$ with $c<\overline{c}$, namely the small spheres. One of the interesting but challenging problems in the geometry of submanifolds is to determine the necessary and sufficient conditions for a compact Riemannian hypersurface in a space form to be spherical. This investigation has been extensively discussed by several mathematicians. In [1], several generalized integral formulas of Minkowski type were derived for compact Riemannian hypersurfaces in both Riemannian and Lorentzian manifolds in the presence of an arbitrary vector field. When the vector field has a specific form, such as being Killing or conformal, these integral formulas have been utilized to characterize totally umbilical compact Riemannian hypersurfaces in both Riemannian and Lorentzian manifolds. The study of spacelike hypersurfaces with constant mean curvature in Lorentz spaces, including de Sitter and anti-de Sitter spaces, has also been explored in various works (see, for example, [2,3,4]). Additional studies can be found in [5,6,7,8,9]. In particular, a natural question arises regarding the consideration of compact spacelike hypersurfaces in $S_1^{n+1}\left(\overline{c}\right)$ without the constancy condition on the mean curvature or the scalar curvature. For a more comprehensive understanding of hyperbolic cylinders in anti-de Sitter space, totally umbilical hypersurfaces in de Sitter space, and isoparametric hypersurfaces in Lorentzian space forms, we refer to [10,11,12], as well as their respective references.

This article investigates the characterization of compact spacelike hypersurfaces in de Sitter space without assuming constant mean curvature. It provides necessary and sufficient conditions for such hypersurfaces to be spheres, based on Ricci curvature constraints. The motivations for characterizing spheres in de Sitter space are as follows: First, de Sitter spacetime models a universe with a positive cosmological constant, and studying spheres with specific Ricci conditions can help identify other geometric structures. Second, examining Ricci curvature in a specific direction deepens our understanding of its impact on the geometric and physical properties of spacelike hypersurfaces. Third, restricting Ricci curvature may provide new geometric insights and lead to novel characterizations of spacelike hypersurfaces in Lorentzian manifolds.

In Section 2, we state preliminaries and some basic concepts and fundamental formulas in the theory of hypersurfaces in a de Sitter space (cf. [13,14,15]). In Section 3, we provide a brief overview of spheres in de Sitter space. In Section 4, we present the main result (Theorem 2), which characterizes spheres in $S_1^{n+1}\left(\overline{c}\right)$ by utilizing the Ricci curvature of the induced vector field on a compact spacelike hypersurface M in $S_1^{n+1}\left(\overline{c}\right).$ In Section 5, we give a characterization of spheres of de Sitter space, viewed as a Robertson–Walker space.

2. Preliminaries

Consider the de Sitter $S_1^{n+1}\left(\overline{c}\right)$ as the hypersurface of the Minkowski space ${\mathbb{R}}_1^{n+2}$ defined in the previous section. Define the spacelike position vector field $\overline{\psi }:S_1^{n+1}\left(\overline{c}\right)⟶{\mathbb{R}}_1^{n+2}$ with $\overline{\psi }\left(X\right)=X$. Then, it has a shape operator $\overline{A}=-\sqrt{\overline{c}}I$ with respect to the spacelike unit normal vector field $\overline{N}=\sqrt{\overline{c}}$$\overline{\psi },$ where I denotes the identity operator.

Consider a compact connected n-dimensional spacelike hypersurface $\left(M,g\right)$ in the de Sitter space $\left(S_1^{n+1}\left(\overline{c}\right),\overline{g}\right).$ A vector field X in $S_1^{n+1}$ is called spacelike if $g\left(X,X\right)>0$, timelike if $g\left(X,X\right)<0$, and lightlike or null if $g\left(X,X\right)=0$ everywhere. $\nabla ,\overline{\nabla }$ and D denote the Levi–Civita connections of $M,$$S_1^{n+1}\left(\overline{c}\right)$ and ${\mathbb{R}}_1^{n+2}$, respectively. Let $\mathfrak{X}\left(M\right)$ and $\mathfrak{X}\left(S_1^{n+1}\left(\overline{c}\right)\right)$ denote the sets of all smooth vector fields on M and $S_1^{n+1}\left(\overline{c}\right)$, respectively. Then, the Gauss and Weingarten formulas for $S_1^{n+1}\left(\overline{c}\right)$ in ${\mathbb{R}}_1^{n+2}$ are given by

$$D_{X}Y={\overline{\nabla }}_{X}Y+\overline{g}\left(\overline{A}\left(X\right),Y\right)\overline{N},$$

(1)

$$D_X\overline{N}=-\overline{A}\left(X\right)=\sqrt{\overline{c}}X,$$

(2)

for all $X,Y\in \mathfrak{X}\left(S_1^{n+1}\left(\overline{c}\right)\right).$

Recall that a smooth vector field Z on a pseudo-Riemannian manifold $\left(\tilde{M},\tilde{g}\right)$ is said to be conformal if the Lie derivative $L_Z$ of $\tilde{g}$ with respect to Z satisfies

$$\begin{array}{c}\hfill L_Z\tilde{g}\left(X,Y\right)=2\phi \tilde{g}\left(X,Y\right),\end{array}$$

for all $X,Y\in \mathfrak{X}\left(\tilde{M}\right),$ where $\phi$ is a smooth function on $\tilde{M},$ called the potential function of Z. In addition, we say that Z is closed conformal if there exists a smooth function $\phi$ on $\tilde{M}$ such that

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{X}Z=\phi X,\end{array}$$

for all $X\in X\left(\tilde{M}\right)$, where $\tilde{\nabla }$ is the Levi–Civita connections of $\tilde{M}.$

Closed conformal vector fields are also referred to as concircular vector fields, and are called concurrent vector fields when $\phi =1$.

Recall that every compact spacelike hypersurface M in $S_1^{n+1}\left(\overline{c}\right)$ is diffeomorphic to an n-sphere [13], and hence is orientable. Therefore, there exists a timelike unit normal vector field globally defined on M. Let N be a timelike unit normal vector field on M. Then, the Gauss and Weingarten formulas for M in $S_1^{n+1}\left(\overline{c}\right)$ are given by

$${\overline{\nabla }}_{X}Y={\nabla }_{X}Y-g\left(A\left(X\right),Y\right)N,$$

(3)

$${\overline{\nabla }}_{X}N=-A\left(X\right),$$

(4)

for all $X,Y\in \mathfrak{X}\left(M\right),$ where A is the shape operator of M in $S_1^{n+1}\left(\overline{c}\right)$ derived from the timelike unit normal vector field N.

The Ricci curvature $Ric$ of M can be expressed as

$$Ric\left(X,Y\right)=\left(n-1\right)\overline{c}\overline{g}\left(X,Y\right)+nHg\left(A\left(X\right),Y\right)+g\left(A\left(X\right),A\left(Y\right)\right),$$

(5)

where H is the mean curvature of M, defined by

$$H=-{\displaystyle \frac{1}{n}}trace\left(A\right).$$

(6)

Since the de Sitter space is of constant curvature, the Codazzi equation for M has the form

$$\left(\nabla A\right)\left(X,Y\right)=\left(\nabla A\right)\left(Y,X\right),$$

(7)

where $X,Y\in \mathfrak{X}\left(M\right)$ and

$$\left(\nabla A\right)\left(X,Y\right)={\nabla }_{X}A\left(Y\right)-A\left({\nabla }_{X}Y\right).$$

(8)

By using (7) and the symmetry of the shape operator A, the mean curvature H has the gradient

$$\nabla H=-{\displaystyle \frac{1}{n}}\sum _i\left(\nabla A\right)\left(e_i,e_i\right),$$

(9)

where $\left\{e_1,e_2,...,e_n\right\}$ is a local orthonormal frame on $M.$

For a nonzero parallel (constant) vector field $\overline{Z}$ on Minkowski space ${\mathbb{R}}_1^{n+2},$ we denote the restriction of $\overline{Z}$ to $S_1^{n+1}\left(\overline{c}\right)$ by Z. Then, we denote by $\overline{\sigma }$ the smooth function on $S_1^{n+1}\left(\overline{c}\right)$, called the support function, which is defined by $\overline{\sigma }=〈Z,\overline{N}〉$, where $〈·,·〉$ is the Lorentzian metric on ${\mathbb{R}}_1^{n+2}.$ If $Z^{\top }$ is the tangential component of Z to $S_1^{n+1}\left(\overline{c}\right),$ then Z can be expressed as

$$\begin{array}{c}\hfill Z=Z^{\top }+\overline{\sigma }\overline{N}.\end{array}$$

Note that $Z^{\top }$ is timelike in the case where Z is timelike or lightlike, and it could be timelike, spacelike, or lightlike in the case where Z is spacelike.

Since $D_{X}Z=0$ for all $X\in \mathfrak{X}\left(S_1^{n+1}\left(\overline{c}\right)\right)$, we can deduce from (1) and (2) that

$${\overline{\nabla }}_X{Z}^{\top }=-\sqrt{\overline{c}}\overline{\sigma }X,\nabla \overline{\sigma }=\sqrt{\overline{c}}{Z}^{\top },$$

(10)

for all $X\in \mathfrak{X}\left(S_1^{n+1}\left(\overline{c}\right)\right).$

It follows from (10) that $Z^{\top }$ is a closed conformal (timelike) vector field on $S_1^{n+1}\left(\overline{c}\right)$ with potential function $-\sqrt{\overline{c}}$ $\overline{\sigma }$. If the restriction of $Z^{\top }$ to the hypersurface M is denoted by $\xi ,$ then we have

$$\xi ={\xi }^{\top }-\theta N,$$

(11)

where ${\xi }^{\top }$ is the tangential component of $\xi$ (called the induced vector field on M) and $\theta =\overline{g}(\xi ,N)$ is the support function on $M,$ where N is a timelike unit vector field normal to M that can be chosen so that $\theta =\overline{g}(\xi ,N)<0.$ Indeed, it is evident that when the manifold is Lorentzian (in this case, the de Sitter space), the reversed Cauchy–Schwarz inequality gives

$$\theta \le -\sqrt{-\overline{g}\left(\xi ,\xi \right)}$$

(12)

Now, we utilize $\sigma$ to denote the restriction of the support function $\overline{\sigma }$ to the hypersurface $M.$ We call the function $\sigma$ the associated function of the hypersurface. Then, the tangential and the normal components of the gradient $\nabla \overline{\sigma }$ can be, respectively, expressed as

$${\left(\nabla \overline{\sigma }\right)}^{\top }=\nabla \sigma =\sqrt{\overline{c}}{\xi }^{\top },$$

(13)

$$\begin{array}{c}\hfill {\left(\nabla \overline{\sigma }\right)}^{\perp }=-\sqrt{\overline{c}}\theta N.\end{array}$$

Taking the covariant derivative in (11) with respect to $X\in \mathfrak{X}\left(M\right),$ and using (3) and (4), we obtain

$${\nabla }_X{\xi }^{\top }=-\sqrt{\overline{c}}\sigma X-\theta A\left(X\right),\nabla \theta =-A\left({\xi }^{\top }\right).$$

(14)

It follows, by (14), that the divergence of ${\xi }^{\top }$ is given by

$$div\left({\xi }^{\top }\right)=-n\left(\sqrt{\overline{c}}\sigma -\theta H\right).$$

(15)

A hypersurface $\left(M,g\right)$ of the de Sitter $S_1^{n+1}\left(\overline{c}\right)$ is called totally umbilical if its shape operator A is a multiple of the identity, that is, $A=\Phi I,$ where $\Phi$ is a scalar. In particular, M is called totally geodesic if A is identically zero (cf. [15]).

3. Types of Spheres in de Sitter Space

Before presenting the main results, we will first provide a brief overview of spheres in de Sitter space. In this context, spheres refer to spacelike hypersurfaces that are locally spherical and arise as intersections between the de Sitter space and Euclidean spheres in the ambient Minkowski space.

Consider, as above, the de Sitter space $S_1^{n+1}\left(\overline{c}\right)$ as a hypersurface of the Minkowski space ${\mathbb{R}}_1^{n+2}.$ For $c<\overline{c},$ the small sphere $S^n\left(c\right)$ in $S_1^{n+1}\left(\overline{c}\right)$ can be defined as

$$\begin{array}{c}\hfill S^n\left(c\right)=\left\{\left(Y_1,\cdots ,Y_{n+2}\right)\in {\mathbb{R}}_1^{n+2}:-Y_1^2+\sum _{i=2}^{n+2}{Y}_i^2={\displaystyle \frac{1}{\overline{c}}},Y_1=\sqrt{{\displaystyle \frac{1}{c}}-{\displaystyle \frac{1}{\overline{c}}}}\right\}.\end{array}$$

Consequently, $S^n\left(c\right)$ is a hypersurface of the de Sitter space $S_1^{n+1}\left(\overline{c}\right)$, with a timelike unit normal vector field $N_s$ of the form

$$\begin{array}{c}\hfill N_s\left(Y_1,\cdots ,Y_{n+2}\right)=\left(-\sqrt{{\displaystyle \frac{\overline{c}}{c}}},\sqrt{\overline{c}-c}{Y}_2,\cdots ,\sqrt{\overline{c}-c}{Y}_{n+2}\right),\end{array}$$

for all $\left(Y_1,\cdots ,Y_{n+2}\right)\in S^n\left(c\right).$

Let $\overline{\nabla }$ and ∇ denote the Levi–Civita connections of $S_1^{n+1}\left(\overline{c}\right)$ and $S^n\left(c\right)$, respectively. Then, we have

$$\begin{array}{c}\hfill {\overline{\nabla }}_X{N}_s=\sqrt{\overline{c}-c}X,\end{array}$$

for all $X\in \mathfrak{X}\left(S^n\left(c\right)\right).$

Hence, the shape operator of the hypersurface $S^n\left(c\right)$ is given by

$$\begin{array}{c}\hfill A=-\sqrt{\overline{c}-c}I=-HI,\end{array}$$

where H is the mean curvature of the hypersurface $S^n\left(c\right).$

Since $c<\overline{c}$, H is a nonzero constant, which means that the small sphere $S^n\left(c\right)$ is a totally umbilical submanifold of the de Sitter space $S_1^{n+1}\left(\overline{c}\right),$ although it is not totally geodesic. A similar result concerning small spheres in Riemannian geometry was given in [16].

Now, denote the tangential component of the vector field $\xi$ onto the small sphere $S^n\left(c\right)$ as ${\xi }^{\top }$, and define $\theta =\overline{g}\left(\xi ,N_s\right)$. Then, we have

$$\begin{array}{c}\hfill \xi ={\xi }^{\top }-\theta N_s,\end{array}$$

and by using the definitions of $\xi$ and $N_s,$ we can easily see that

$$\begin{array}{c}\hfill \overline{g}\left(\xi ,N_s\right)=\sqrt{{\displaystyle \frac{\overline{c}-c}{\overline{c}}}}\sigma ,\end{array}$$

and consequently,

$$\theta ={\displaystyle \frac{1}{\sqrt{\overline{c}}}}H\sigma ,$$

(16)

where $\sigma$ is the restriction of $\overline{\sigma }$ to $S^n\left(c\right).$

By (14) and (16), we obtain

$$\begin{array}{c}\hfill {\nabla }_X{\xi }^{\top }=-{\displaystyle \frac{\left(\overline{c}-H^2\right)}{\sqrt{\overline{c}}}}\sigma X,\nabla \theta =H{\xi }^{\top },\end{array}$$

for all $X\in \mathfrak{X}\left(S^n\left(c\right)\right)$.

It follows that the divergence of ${\xi }^{\top }$ is given by

$$div\left({\xi }^{\top }\right)=-{\displaystyle \frac{n}{\sqrt{\overline{c}}}}\left(\overline{c}-H^2\right)\sigma .$$

(17)

The Ricci curvature of the small sphere $S^n\left(c\right)$ is given by

$$Ric\left(X,Y\right)=\left(n-1\right)\left(\overline{c}-H^2\right)g\left(X,Y\right),$$

(18)

for all $X,Y\in \mathfrak{X}\left(S^n\left(c\right)\right)$.

4. Characterizations of Spheres in de Sitter Space

Consider the de Sitter space $S_1^{n+1}\left(\overline{c}\right),$$n\ge 2$, and let M be a compact spacelike hypersurface with mean curvature H, induced vector field ${\xi }^{\top },$ support function $\theta ,$ and associated function $\sigma .$

In this section, we derive an interesting characterization of spheres in $S_1^{n+1}\left(\overline{c}\right)$ using a lower bound on the integral of the Ricci curvature $Ric\left({\xi }^{\top },{\xi }^{\top }\right)$. Many researchers have investigated such characterizations of spheres in the unit sphere $S^{n+1}$ in Euclidean space (see [17,18]), typically using the tangent component of the position vector field on the hypersurface. However, for Riemannian hypersurfaces in spheres and spacelike hypersurfaces in de Sitter spacetime, since the position vector field may change its causal character, we use constant (i.e., parallel) vector fields from the ambient manifold instead.

The following lemmas are crucial for proving the main results.

Lemma 1.

Let $\left(M,g\right)$ be a compact n-dimensional spacelike hypersurface of the de Sitter space $S_1^{n+1}\left(\overline{c}\right),$$n\ge 2$, with mean curvature $H,$ induced vector field ${\xi }^{\top },$ support function $\theta ,$ and associated function $\sigma .$ Then,

$$\theta {\xi }^{\top }\left(H\right)=divH\left(\theta {\xi }^{\top }\right)+Hg\left(A\left({\xi }^{\top }\right),{\xi }^{\top }\right)+n\theta H\left(\sqrt{\overline{c}}\sigma -\theta H\right).$$

(19)

Proof. 

By using (15), we obtain

$$\begin{array}{ccc}\hfill divH\left(\theta {\xi }^{\top }\right)& =& \theta {\xi }^{\top }\left(H\right)+Hdiv\left(\theta {\xi }^{\top }\right)\hfill \\ & =& \theta {\xi }^{\top }\left(H\right)+H\left({\xi }^{\top }\left(\theta \right)+\theta div\left({\xi }^{\top }\right)\right)\hfill \\ & =& \theta {\xi }^{\top }\left(H\right)+H{\xi }^{\top }\left(\theta \right)+\theta Hdiv\left({\xi }^{\top }\right)\hfill \\ & =& \theta {\xi }^{\top }\left(H\right)+Hg\left(\nabla \theta ,{\xi }^{\top }\right)-n\theta H\left(\sqrt{\overline{c}}\sigma -\theta H\right)\hfill \\ & =& \theta {\xi }^{\top }\left(H\right)-Hg\left(A\left({\xi }^{\top }\right),{\xi }^{\top }\right)-n\theta H\left(\sqrt{\overline{c}}\sigma +\theta H\right),\hfill \end{array}$$

which yields the result. □

Lemma 2.

Let $\left(M,g\right)$ be a compact n-dimensional spacelike hypersurface of the de Sitter $S_1^{n+1}\left(\overline{c}\right),$$n\ge 2$, with mean curvature $H,$ induced vector field ${\xi }^{\top },$ support function $\theta ,$ and associated function $\sigma .$ Then,

$$\sqrt{\overline{c}}{|{\xi }^{\top }|}^2=div\left(\sigma {\xi }^{\top }\right)+n\sqrt{\overline{c}}{\sigma }^2-n\sigma \theta H.$$

(20)

Proof. 

By (13), we have

$$\begin{array}{ccc}\hfill div\left(\sigma {\xi }^{\top }\right)& =& {\xi }^{\top }\left(\sigma \right)+\sigma div\left({\xi }^{\top }\right)\hfill \\ & =& g\left(\nabla \sigma ,{\xi }^{\top }\right)-n\sigma \left(\sqrt{\overline{c}}\sigma -\theta H\right)\hfill \\ & =& \sqrt{\overline{c}}{|{\xi }^{\top }|}^2-n\sqrt{\overline{c}}{\sigma }^2+n\sigma \theta H,\hfill \end{array}$$

which yields the result. □

We are now ready to express the Laplacian $\Delta \theta$ of the support function in terms of the norm of the shape operator A of $M.$

Lemma 3.

Let $\left(M,g\right)$ be an n-dimensional spacelike hypersurface of the de Sitter space $S_1^{n+1}\left(\overline{c}\right),$$n\ge 2$, with mean curvature $H,$ induced vector field ${\xi }^{\top },$ and associated function $\sigma .$ Then, the Laplacian of the support function θ is given by

$$\Delta \theta =n{\xi }^{\top }\left(H\right)-n\sqrt{\overline{c}}\sigma H+\theta {\parallel A\parallel }^2,$$

(21)

where ${\parallel A\parallel }^2$ denotes the trace of $A^2$.

Proof. 

Let $\left\{e_1,e_2,\dots ,e_n\right\}$ be a local orthonormal frame of vector fields in $M.$ Taking the divergence for the second equation in (14), we obtain

$$\begin{array}{ccc}\hfill \Delta \theta & =& -div\left(A\left({\xi }^{\top }\right)\right)\hfill \\ & =& -\sum _{i=1}^{n}g\left({\nabla }_{e_i}A\left({\xi }^{\top }\right),e_i\right).\hfill \end{array}$$

Using (8), we obtain

$$\begin{array}{ccc}\hfill \Delta \theta & =& -\sum _{i=1}^{n}g\left(\left({\nabla }_{e_i}A\right){\xi }^{\top },e_i\right)-\sum _{i=1}^{n}g\left(A\left({\nabla }_{e_i}{\xi }^{\top }\right),e_i\right)\hfill \\ & =& -\sum _{i=1}^{n}g\left({\xi }^{\top },\left({\nabla }_{e_i}A\right)e_i\right)-\sum _{i=1}^{n}g\left(A\left({\nabla }_{e_i}{\xi }^{\top }\right),e_i\right),\hfill \end{array}$$

where we have used the fact that since A is symmetric, then so is ${\nabla }_{e_i}A.$

Then, according to (9), and by making use of (6) and (14), we have

$$\begin{array}{ccc}\hfill \Delta \theta & =& ng\left({\xi }^{\top },\nabla H\right)+\sqrt{\overline{c}}\sigma \sum _{i=1}^{n}g\left(A\left(e_i\right),e_i\right)+\theta \sum _{i=1}^{n}g\left(A^2\left(e_i\right),e_i\right)\hfill \\ & =& n{\xi }^{\top }\left(H\right)-n\sqrt{\overline{c}}\sigma H+\theta {\parallel A\parallel }^2,\hfill \end{array}$$

as desired. □

The first result of this paper directly follows from Formula (21).

Theorem 1.

Let $(M,g)$ be a compact n-dimensional spacelike hypersurface of the de Sitter space $S_1^{n+1}\left(\overline{c}\right)$, with $n\ge 2$, that has a constant mean curvature along the integral curves of the induced vector field ${\xi }^{\top }$. Then, $(M,g)$ is isometric to a sphere.

Proof. 

By using (15) and the fact that

$$div\left(H{\xi }^{\top }\right)=Hdiv\left({\xi }^{\top }\right)+{\xi }^{\top }\left(H\right),$$

(22)

Formula (21) becomes

$$\begin{array}{ccc}\hfill \Delta \theta & =& n{\xi }^{\top }\left(H\right)+\theta \left({∥A∥}^2-n{H}^2\right)+nH\left(\theta H-\sqrt{\overline{c}}\sigma \right)\hfill \\ & =& n{\xi }^{\top }\left(H\right)+\theta \left({∥A∥}^2-n{H}^2\right)+Hdiv\left({\xi }^{\top }\right)\hfill \\ & =& (n-1){\xi }^{\top }\left(H\right)+\theta \left({∥A∥}^2-n{H}^2\right)+div\left(H{\xi }^{\top }\right)\hfill \end{array}$$

By integrating both sides of the above equation over M, we obtain

$$\underset{M}{\int }\theta \left({∥A∥}^2-n{H}^2\right)dV=-(n-1)\underset{M}{\int }{\xi }^{\top }\left(H\right)dV$$

Since ${∥A∥}^2\ge n{H}^2$ and $\theta$ is nonzero with a constant sign (as indicated by Inequality (12)), and assuming that H is constant along the integral curves of ${\xi }^{\top }$, it follows from the Cauchy–Schwarz inequality that

$$\begin{array}{c}\hfill \underset{M}{\int }\theta \left({∥A∥}^2-n{H}^2\right)dV=0.\end{array}$$

This leads to the conclusion that ${∥A∥}^2=n{H}^2$. Equality in the Cauchy–Schwarz inequality occurs if and only if

$$A=-HI.$$

(23)

Thus, $(M,g)$ is a totally umbilical hypersurface in $S_1^{n+1}\left(\overline{c}\right)$. According to Lemma 4.35 and Proposition 4.36 in [15], $(M,g)$ is isometric to a sphere $S^n(\overline{c}-H^2)$. □

Corollary 1.

Let $(M,g)$ be a constant mean curvature compact n-dimensional spacelike hypersurface of the de Sitter space $S_1^{n+1}\left(\overline{c}\right)$, with $n\ge 2$. Then, $(M,g)$ is isometric to a sphere.

Our second result outlines the conditions that a spacelike hypersurface in de Sitter space must meet to be classified as a sphere.

Theorem 2.

Let $\left(M,g\right)$ be a compact n-dimensional spacelike hypersurface of the de Sitter space $S_1^{n+1}\left(\overline{c}\right),$$n\ge 2,$ with induced vector field ${\xi }^{\top }$. Then,

$$\underset{M}{\int }Ric\left({\xi }^{\top },{\xi }^{\top }\right)dV\ge {\displaystyle \frac{\left(n-1\right)}{n}}\underset{M}{\int }{\left(div\left({\xi }^{\top }\right)\right)}^{2}dV$$

(24)

if and only if $\left(M,g\right)$ is isometric to a sphere.

Proof. 

Suppose that the Ricci curvature and the induced vector field ${\xi }^{\top }$ on M satisfy (24). By multiplying both sides of (21) by $\theta ,$ we obtain

$$\begin{array}{c}\hfill \theta \Delta \theta =n\theta {\xi }^{\top }\left(H\right)-n\sqrt{\overline{c}}\sigma \theta H+{\theta }^2{∥A∥}^2,\end{array}$$

that is,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\Delta {\theta }^2-{|\nabla \theta |}^2=n\theta {\xi }^{\top }\left(H\right)-n\sqrt{\overline{c}}\sigma \theta H+{\theta }^2{∥A∥}^2.\end{array}$$

By using (14) and (19), we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{2}}\Delta {\theta }^2-{∥A∥}^2& =& ndivH\left(\theta {\xi }^{\top }\right)+nHg\left(A\left({\xi }^{\top }\right),{\xi }^{\top }\right)+n^2\theta H\left(\sqrt{\overline{c}}\sigma -\theta H\right)\hfill \\ & & -n\sqrt{\overline{c}}\sigma \theta H+{\theta }^2{∥A∥}^2,\hfill \end{array}$$

or equivalently

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{2}}\Delta {\theta }^2& =& ndivH\left(\theta {\xi }^{\top }\right)+{∥A∥}^2+nHg\left(A\left({\xi }^{\top }\right),{\xi }^{\top }\right)\hfill \\ & & +n\left(n-1\right)\left(\sqrt{\overline{c}}\sigma \theta H-{\theta }^2{H}^2\right)+{\theta }^2\left({∥A∥}^2-n{H}^2\right).\hfill \end{array}$$

Then, (5) yields

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{2}}\Delta {\theta }^2& =& ndivH\left(\theta {\xi }^{\top }\right)+Ric\left({\xi }^{\top },{\xi }^{\top }\right)-\left(n-1\right)\overline{c}{\left|{\xi }^{\top }\right|}^2\hfill \\ & & +n\left(n-1\right)\left(\sqrt{\overline{c}}\sigma \theta H-{\theta }^2{H}^2\right)+{\theta }^2\left({∥A∥}^2-n{H}^2\right).\hfill \end{array}$$

(25)

Now, by using (15) and (20), we obtain

$$\begin{array}{c}\hfill {\left(div\left({\xi }^{\top }\right)\right)}^2=n\sqrt{\overline{c}}\left(\sqrt{\overline{c}}{\left|{\xi }^{\top }\right|}^2-div\left(\sigma {\xi }^{\top }\right)\right)+n^2\left({\theta }^2{H}^2-\sqrt{\overline{c}}\sigma \theta H\right).\end{array}$$

It follows that

$$\begin{array}{ccc}\hfill n\left(n-1\right)\left({\theta }^2{H}^2-\sqrt{\overline{c}}\sigma \theta H\right)& =& {\displaystyle \frac{\left(n-1\right)}{n}}{\left(div\left({\xi }^{\top }\right)\right)}^2-\left(n-1\right)\overline{c}{\left|{\xi }^{\top }\right|}^2\hfill \\ & & +\left(n-1\right)\sqrt{\overline{c}}div\left(\sigma {\xi }^{\top }\right),\hfill \end{array}$$

(26)

and by substituting this into (25), we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{2}}\Delta {\theta }^2& =& ndiv\left(H\left(\theta {\xi }^{\top }\right)\right)+Ric\left({\xi }^{\top },{\xi }^{\top }\right)-{\displaystyle \frac{\left(n-1\right)}{n}}{\left(div\left({\xi }^{\top }\right)\right)}^2\hfill \\ & & -\left(n-1\right)\sqrt{\overline{c}}div\left(\sigma {\xi }^{\top }\right)+{\theta }^2\left({∥A∥}^2-n{H}^2\right).\hfill \end{array}$$

Since M is compact, integrating both sides of the above equation yields

$$\begin{array}{c}\hfill \underset{M}{\int }\left({\theta }^2\left(n{H}^2-{∥A∥}^2\right)\right)dV=\underset{M}{\int }\left(Ric\left({\xi }^{\top },{\xi }^{\top }\right)-{\displaystyle \frac{\left(n-1\right)}{n}}{\left(div\left({\xi }^{\top }\right)\right)}^2\right)dV.\end{array}$$

Applying inequality (24) to the equation above, it follows that

$$\begin{array}{c}\hfill \underset{M}{\int }{\theta }^2\left(n{H}^2-{∥A∥}^2\right)dV\ge 0.\end{array}$$

Since ${∥A∥}^2\ge n{H}^2,$ by the Cauchy–Schwarz inequality, it follows that

$$\begin{array}{c}\hfill \underset{M}{\int }{\theta }^2\left(n{H}^2-{∥A∥}^2\right)dV=0.\end{array}$$

This, combined with $\theta \ne 0$, leads to the conclusion that ${∥A∥}^2=n{H}^2$. This represents equality in the Cauchy–Schwarz inequality, which occurs if and only if

$$A=-HI.$$

(27)

It follows, by (23), that M is a totally umbilical hypersurface of $S_1^{n+1}\left(\overline{c}\right)$. Hence, according to Lemma 4.35 and Proposition 4.36 in [15], $(M,g)$ is isometric to a sphere $S^n\left(\overline{c}-H^2\right).$

Conversely, assume that M is a totally umbilical hypersurface which is isometric to the sphere $S^n\left(\overline{c}-H^2\right).$ It follows, by (18), that the Ricci curvature of ${\xi }^{\top }$ is given by

$$Ric\left({\xi }^{\top },{\xi }^{\top }\right)=\left(n-1\right)\left(\overline{c}-H^2\right){\left|{\xi }^{\top }\right|}^2.$$

(28)

Since M is compact, then integrating (20) yields

$$\underset{M}{\int }{\left|{\xi }^{\top }\right|}^{2}dV={\displaystyle \frac{n}{\overline{c}}}\underset{M}{\int }\left(\overline{c}-H^2\right){\sigma }^{2}dV.$$

(29)

By integrating the both sides of (28), and making use of (29), we obtain

$$\underset{M}{\int }Ric\left({\xi }^{\top },{\xi }^{\top }\right)dV=\left(n-1\right)\underset{M}{\int }{\displaystyle \frac{n}{\overline{c}}}{\left(\overline{c}-H^2\right)}^2{\sigma }^{2}dV.$$

(30)

By using (17), we have

$$\begin{array}{c}\hfill {\left(div\left({\xi }^{\top }\right)\right)}^2={\displaystyle \frac{n^2}{\overline{c}}}{\left(\overline{c}-H^2\right)}^2{\sigma }^2.\end{array}$$

Substituting this into (30), we obtain

$$\begin{array}{c}\hfill \underset{M}{\int }Ric\left({\xi }^{\top },{\xi }^{\top }\right)dV={\displaystyle \frac{\left(n-1\right)}{n}}\underset{M}{\int }{\left(div\left({\xi }^{\top }\right)\right)}^{2}dV,\end{array}$$

and hence, inequality (24) holds. □

The following result is a different formulation of Theorem 2.

Theorem 3.

Let $\left(M,g\right)$ be a compact n-dimensional spacelike hypersurface of the de Sitter space $S_1^{n+1}\left(\overline{c}\right)$, $n\ge 2$, with induced vector field ${\xi }^{\top }$ and support function θ. Then,

$$\begin{array}{c}\hfill \underset{M}{\int }Ric\left({\xi }^{\top },{\xi }^{\top }\right)\ge \underset{M}{\int }\left(n-1\right)\left[n\left({\theta }^2{H}^2-\overline{c}{\sigma }^2\right)+2\overline{c}{\left|{\xi }^{\top }\right|}^2\right]\end{array}$$

(31)

if and only if M is isometric to a sphere $S^n\left(c\right)$ of radius $c^{-1}$, with $c=\overline{c}-H^2$.

Proof. 

By combining the two Formulas (20) and (25), we deduce that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{2}}\Delta {\theta }^2& =& ndiv\left(H\left(\theta {\xi }^{\top }\right)\right)+Ric\left({\xi }^{\top },{\xi }^{\top }\right)-2\left(n-1\right)\overline{c}{\left|{\xi }^{\top }\right|}^2+{\theta }^2\left({∥A∥}^2-n{H}^2\right)\hfill \\ & & +\left(n-1\right)div\left(\sigma {\xi }^{\top }\right)+n\left(n-1\right)\left(\overline{c}{\sigma }^2-{\theta }^2{H}^2\right).\hfill \end{array}$$

Integrating both sides of the above equation and recalling that M is compact, we obtain

$$\begin{array}{ccc}\hfill {\theta }^2\underset{M}{\int }\left({∥A∥}^2-n{H}^2\right)dV& =& \underset{M}{\int }\left(n-1\right)\left[n\left({\theta }^2{H}^2-\overline{c}{\sigma }^2\right)+2\overline{c}{\left|{\xi }^{\top }\right|}^2\right]dV\hfill \\ & & -\underset{M}{\int }Ric\left({\xi }^{\top },{\xi }^{\top }\right)dV.\hfill \end{array}$$

(32)

Given that ${∥A∥}^2-n{H}^2\ge 0$, by the Cauchy–Schwarz inequality, and the assumption (31), we deduce from (32) that ${∥A∥}^2=n{H}^2$, meaning that $\left(M,g\right)$ is totally umbilical. According to Lemma 4.35 and Proposition 4.36 in [15], the mean curvature H is constant, and $(M,g)$ is isometric to a sphere $S^n\left(\overline{c}-H^2\right).$

The converse can be established in a similar manner to the proof of Theorem 2. □

Example 1.

Consider the small sphere $S^n\left(c\right)$ introduced in Section 3. From (17) and (18), we obtain

$$Ric\left({\xi }^{\top },{\xi }^{\top }\right)-{\displaystyle \frac{\left(n-1\right)}{n}}{\left(div\left({\xi }^{\top }\right)\right)}^2=(n-1)c{|{\xi }^{\top }|}^2-n\left(n-1\right)\left({\displaystyle \frac{c^2}{\overline{c}}}{\sigma }^2\right),$$

which simplifies to

$$Ric\left({\xi }^{\top },{\xi }^{\top }\right)-{\displaystyle \frac{\left(n-1\right)}{n}}{\left(div\left({\xi }^{\top }\right)\right)}^2=\left(n-1\right)c\left({|{\xi }^{\top }|}^2-{\displaystyle \frac{nc}{\overline{c}}}{\sigma }^2\right).$$

By applying the relation

$$div\left(\sigma {\xi }^{\top }\right)=\sigma div\left({\xi }^{\top }\right)+{\xi }^{\top }\sigma ,$$

along with (13) and (17), we obtain

$$\begin{array}{c}\hfill div\left(\sigma {\xi }^{\top }\right)=\sigma \left({\displaystyle \frac{-nc}{\sqrt{\overline{c}}}}\sigma \right)+g\left(\nabla \sigma ,{\xi }^{\top }\right)={\displaystyle \frac{-nc}{\sqrt{\overline{c}}}}{\sigma }^2+\sqrt{\overline{c}}{|{\xi }^{\top }|}^2,\end{array}$$

which can be rewritten as

$$div\left(\sigma {\xi }^{\top }\right)=\sqrt{\overline{c}}\left({|{\xi }^{\top }|}^2-{\displaystyle \frac{nc}{\overline{c}}}{\sigma }^2\right).$$

Therefore,

$$\begin{array}{c}\hfill Ric\left({\xi }^{\top },{\xi }^{\top }\right)-{\displaystyle \frac{\left(n-1\right)}{n}}{\left(div\left({\xi }^{\top }\right)\right)}^2=(n-1){\displaystyle \frac{c}{\sqrt{\overline{c}}}}div\left(\sigma {\xi }^{\top }\right),\end{array}$$

which yields

$$\begin{array}{c}\hfill \underset{M}{\int }Ric\left({\xi }^{\top },{\xi }^{\top }\right)dV={\displaystyle \frac{\left(n-1\right)}{n}}\underset{M}{\int }{\left(div\left({\xi }^{\top }\right)\right)}^{2}dV.\end{array}$$

This shows that the small sphere $S^n\left(c\right)$ is an example of a compact spacelike hypersurface in de Sitter space that satisfies the Ricci curvature condition (24) stated in Theorem 2.

Remark 1.

We note that spheres are not present in Minkowski spaces or anti-de Sitter spaces (see [19]). However, in general Lorentzian manifolds, we can discuss the existence of extrinsic spheres, which are totally umbilical hypersurfaces with nonzero constant mean curvatures.

Remark 2.

A similar result to Theorem 3 was provided in [17] for the case where $\left(M,g\right)$ is a Riemannian hypersurface of the unit sphere $S^{n+1}$.

5. Spacelike Hypersurfaces in de Sitter Space Viewed as a Robertson–Walker (RW) Spacetime

As mentioned in Section 1, assuming $n\ge 2$, de Sitter space $\left(S_1^{n+1}\left(\overline{c}\right),\overline{g}\right)$ is the unique geodesically complete, simply connected spacetime with constant positive curvature $\overline{c}$. If $\left(S^n,d{s}^2\right)$ represents the standard unit sphere (i.e., the round Riemannian sphere with constant curvature $=1$), de Sitter space can be expressed in global coordinates as the Lorentzian warped product

$$\left(\mathbb{R}\times S^n,-d{t}^2+{\displaystyle \frac{1}{\overline{c}}}{cosh}^2\left(\sqrt{\overline{c}}t\right)d{s}^2\right).$$

In other words, de Sitter space can be viewed as a Robertson–Walker (RW) spacetime. In fact, RW spacetimes are well known for their importance in physics, especially in the theory of general relativity.

The vector field ${\partial }_t$ is a timelike unit constant vector field in Minkowski space ${\mathbb{R}}_1^{n+2}$, and the vector filed $\xi ={\displaystyle \frac{1}{\overline{c}}}{cosh}^2\left(\sqrt{\overline{c}}t\right){\partial }_t$ can be shown to be a closed conformal vector field in ${\mathbb{R}}_1^{n+2}$, satisfying

$${\overline{\nabla }}_X\xi ={\displaystyle \frac{1}{\sqrt{\overline{c}}}}sinh\left(2\sqrt{\overline{c}}t\right)X,$$

for all vector fields $X\in \mathfrak{X}\left({\mathbb{R}}_1^{n+2}\right)$, where $\overline{\nabla }$ denotes the Levi–Civita connection of ${\mathbb{R}}_1^{n+2}$, and $\mathfrak{X}\left({\mathbb{R}}_1^{n+2}\right)$ is the space of all vector fields in ${\mathbb{R}}_1^{n+2}$.

Let $(M,g)$ be a spacelike hypersurface in $\left(S_1^{n+1}\left(\overline{c}\right),\overline{g}\right)$. Since ${\partial }_t$ establishes an orientation for $S_1^{n+1}$, let N be a globally defined unit normal vector field to M. We define $\Theta$, the support function of a ${\partial }_t$, as a smooth function on M given by $\Theta =\overline{g}\left({\partial }_t,N\right)$. Referring to the notation introduced in the middle of Section 2, we observe that if $\theta =\overline{g}\left(\xi ,N\right)$, then $\theta ={\displaystyle \frac{1}{\overline{c}}}{cosh}^2\left(\sqrt{\overline{c}}t\right)\Theta$. Since $\overline{g}({\partial }_t,{\partial }_t)=-1$, then by applying (12) to $\xi$, we obtain

$$\Theta \le -1$$

(33)

If ${\partial }_t^{\top }$ denotes the component of ${\partial }_t$ that is tangent to M, then it can be expressed as

$${\partial }_t^{\top }={\partial }_t+\Theta N.$$

Consider the function $h:M\to \mathbb{R}$ (called the height function) defined by $h\left(x\right)=(\pi \circ \psi )\left(x\right)$, where $\psi :M\to S_1^{n+1}\left(\overline{c}\right)$ is the isometric immersion representing $(M,g)$ as a spacelike hypersurface of $\left(S_1^{n+1}\left(\overline{c}\right),\overline{g}\right)$, and $\pi :S_1^{n+1}\left(\overline{c}\right)\to \mathbb{R}$ is the projection on the base $\mathbb{R}$.

It is clear that the gradient $\overline{\nabla }\pi$ of the projection $\pi$ satisfies $\overline{\nabla }\pi =-{\partial }_t$. It follows that the gradient of h is given by

$$\nabla h=-{\partial }_t^{\top },$$

(34)

The next two formulas, which describe the norm (i.e., length) and Laplacian of h, are established in [20].

$$\begin{array}{cc}\hfill {|\nabla h|}^2& =({\Theta }^2-1),\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill \Delta h& =-2(n-1+{\Theta }^2)\sqrt{\overline{c}}tanh\left(\sqrt{\overline{c}}t\right)-n\Theta H.\hfill \end{array}$$

(36)

The following result provides a characterization of spheres in $\left(S_1^{n+1}\left(\overline{c}\right),\overline{g}\right)$ through an inequality similar to (24) involving the height function h.

Theorem 4.

Let $(M,g)$ be a compact spacelike hypersurface in $\left(S_1^{n+1}\left(\overline{c}\right),\overline{g}\right)$. Then, $(M,g)$ is isometric to a sphere if and only if the height function h satisfies

$$\begin{array}{c}\hfill Ric(\nabla h,\nabla h)\ge \left({\displaystyle \frac{n-1}{n}}\right){\left(\Delta h+2\sqrt{\overline{c}}tanh\left(\sqrt{\overline{c}}t\right){|\nabla h|}^2\right)}^2\end{array}$$

(37)

Proof. 

It is clear that the gradient $\overline{\nabla }\pi$ of the projection $\pi :S_1^{n+1}\to \mathbb{R}$ satisfies $\overline{\nabla }\pi =-{\partial }_t$. As a result, $\nabla h=-{\partial }_t^{\top }$. Therefore, for $f\left(t\right)={\displaystyle \frac{1}{\overline{c}}}{cosh}^2\left(\sqrt{\overline{c}}t\right)$, since $\nabla f=-f^{\prime }{\partial }_t^{\top }=f^{\prime }\nabla h$, the inequality (24), with ${\xi }^{\top }=f{\partial }_t^{\top }$ becomes

$$\begin{array}{cc}\hfill f^{2}Ric\left(\nabla h,\nabla h\right)& \ge \left({\displaystyle \frac{n-1}{n}}\right){\left(\mathrm{div}\left(f\nabla h\right)\right)}^2\hfill \\ \hfill & =\left({\displaystyle \frac{n-1}{n}}\right){\left(f\Delta h+\nabla h\left(f\right)\right)}^2\hfill \\ \hfill & =\left({\displaystyle \frac{n-1}{n}}\right)f^2{\left(\Delta h+{\displaystyle \frac{f^{\prime }}{f}}{|\nabla h|}^2\right)}^2,\hfill \end{array}$$

which leads to (37), and Theorem 4 then follows from Theorem 2. □

We conclude with the following result, which can be obtained from Theorem 4 by examining (34), (35), and (36).

Corollary 2.

Let $(M,g)$ be a compact spacelike hypersurface in $\left(S_1^{n+1}\left(\overline{c}\right),\overline{g}\right)$. Then, $(M,g)$ is isometric to a sphere if and only if

$$\begin{array}{c}\hfill Ric\left({\partial }_t^{\top },{\partial }_t^{\top }\right)\ge n(n-1){\left(2\sqrt{\overline{c}}tanh\left(\sqrt{\overline{c}}t\right)+\Theta H\right)}^2\end{array}$$

6. Conclusions

In conclusion, we emphasize that while spheres in Euclidean geometry are typically characterized by their mean or full Ricci curvature, many descriptions of spheres in de Sitter space depend on assumptions about mean curvature, scalar curvature, or Ricci curvature. For instance, Reference [2] assumes constant mean curvature. In contrast, this paper introduces a new approach by characterizing spheres in de Sitter spacetime using a condition focused on Ricci curvature in a specific direction, rather than considering the full Ricci curvature. This approach is motivated by several key reasons: First, de Sitter spacetime represents a universe with a positive cosmological constant, and by applying specific Ricci curvature conditions, we can create more precise criteria for identifying spheres and other geometric structures within this context. Second, concentrating on Ricci curvature in a particular direction contributes to a better understanding of how directional curvature impacts the geometric and physical properties of spacelike hypersurfaces in de Sitter spacetime. Third, restricting the Ricci curvature to a specific direction may reveal new geometric results that are not evident when examining the full Ricci curvature, offering the potential for innovative characterizations and classifications of spacelike hypersurfaces in Lorentzian manifolds. Finally, our characterization of spheres within the spacelike hypersurfaces of de Sitter space, viewed as a Robertson–Walker space, offers a valuable approach for studying the properties of spacelike hypersurfaces in de Sitter space and determining the conditions under which such a hypersurface is a slice (that is of the form $\left\{t_0\right\}\times S^n$). De Sitter space is characterized by a positive cosmological constant that leads to an accelerated exponential expansion of spacetime. By expressing de Sitter space $S_1^{n+1}\left(\overline{c}\right)$ as a generalized Robertson–Walker spacetime (refer to Section 5), this cosmological constant can be calculated in terms of $\overline{c}$. Additionally, analyzing spacelike hypersurfaces (particularly spacelike slices) within this spacetime contributes to a deeper understanding of its structure.