Integral Formulae and Applications for Compact Riemannian Hypersurfaces in Riemannian and Lorentzian Manifolds Admitting Concircular Vector Fields

Abstract

This paper investigates compact Riemannian hypersurfaces immersed in $(n+1)$-dimensional Riemannian or Lorentzian manifolds that admit concircular vector fields, also known as closed conformal vector fields (CCVFs). We focus on the support function of the hypersurface, which is defined as the component of the conformal vector field along the unit-normal vector field, and derive an expression for its Laplacian. Using this, we establish integral formulae for hypersurfaces admitting CCVFs. These results are then extended to compact Riemannian hypersurfaces isometrically immersed in Riemannian or Lorentzian manifolds with constant sectional curvatures, highlighting the crucial role of CCVFs in the study of hypersurfaces. We apply these results to provide characterizations of compact Riemannian hypersurfaces in Euclidean space ${\mathbb{R}}^{n+1}$, Euclidean sphere $S^{n+1}$, and de Sitter space $S_1^{n+1}$.

1. Introduction

Conformal vector fields are fundamental objects in the study of the geometry of manifolds and play a significant role in various areas of differential geometry and theoretical physics, particularly within the framework of general relativity. A vector field $\zeta$ on a pseudo-Riemannian manifold $(M,g)$ is called conformal if its Lie derivative satisfies

$$L_{\zeta }g=2\psi g,$$

for a smooth function $\psi :M\to \mathbb{R}$, called the potential function of $\zeta$. Equivalently, in terms of the Levi-Civita connection ∇, this condition can be written as

$$g\left({\nabla }_X\zeta ,Y\right)+g\left(X,{\nabla }_Y\zeta \right)=2\psi g\left(X,Y\right),$$

for all vector fields $X,Y\in \mathfrak{X}\left(M\right)$. We say that $\zeta$ is a Killing vector field if the potential function $\psi =0$ or, equivalently, if the flow of $\zeta$ consists of isometries.

A conformal vector field $\zeta$ is called closed if its dual 1-form is closed. In this case, the covariant derivative of $\zeta$ satisfies

$${\nabla }_X\zeta =\psi X,$$

for all $X\in \mathfrak{X}\left(M\right)$. Vector fields satisfying this condition are also known as concircular vector fields and are specifically called concurrent vector fields when the potential function is a nonzero constant. Throughout this work, we use the term closed conformal vector fields (CCVFs) to denote such vector fields. CCVFs are essential in characterizing and classifying submanifolds, such as hypersurfaces and minimal surfaces. Moreover, these vector fields appear in various geometric contexts. For example, in Euclidean space $\left({\mathbb{R}}^{n+1},〈·,·〉\right)$, the position vector field is a CCVF. Another notable case arises on the hypersphere $S^n\left(c\right)$ with constant curvature c. Let N be the unit-normal vector field on $S^n\left(c\right)$ embedded in ${\mathbb{R}}^{n+1}$. Let Z be a constant vector field in ${\mathbb{R}}^{n+1}$. Then, its restriction to the hypersphere $S^n\left(c\right)$ can be written as $Z=\zeta +\rho N$, where $\zeta$ is tangent to $S^n\left(c\right)$, and $\rho =\langle Z,N\rangle$ defines a smooth function. It follows that $\zeta$ is a CCVF on $S^n\left(c\right)$ with potential function $-\sqrt{c}\rho$. These examples demonstrate the significance of CCVFs in understanding the geometric properties of hypersurfaces.

The study of conformal and closed conformal vector fields (CCVFs) has been a topic of significant interest in differential geometry. Various researchers have developed integral formulae for hypersurfaces in Riemannian manifolds that admit such vector fields. A foundational contribution was made by Chern [1] in 1959, who introduced integral formulae for hypersurfaces in Euclidean space, forming the basis for studying geometric properties and uniqueness theorems. In 1965, Otsuki [2] extended these formulae to hypersurfaces in Riemannian manifolds, generalizing classical results to spaces with conformal vector fields. Later, in 1971, Chen [3] derived additional integral formulae for closed hypersurfaces in Euclidean spaces. These formulae have been applied to the study of total curvature and the topology of hypersurfaces, offering new insights into their geometric structure. In 1992, Deshmukh [4] derived integral formulae for compact hypersuforfaces in Euclidean space ${\mathbb{R}}^{n+1}$ in terms of scalar and Ricci curvatures. These results were later extended by Vlachos [5] to non-flat space forms, such as the sphere $S^{n+1}$ and the hyperbolic space ${\mathbb{H}}^{n+1}$. Additionally, Alias [6] derived the related formulae for compact hypersurfaces in non-flat space forms. In 2003, Aledo and Gálvez [7] established an integral expression that includes the Ricci and scalar curvatures of compact spacelike hypersurfaces in spacetimes admitting a timelike CCVF. In the same year, Alias, Brasil Jr., and Colares [8] formulated integral expressions for compact spacelike hypersurfaces within conformally stationary spacetimes, which are Lorentzian manifolds that admit a timelike conformal vector field. These formulae provided a framework for analyzing the relationship between the mean curvature and the umbilicity of compact spacelike hypersurfaces (see also [9,10]). Subsequently, in 2010, Albujer, Aledo, and Alias [11] derived an integral formula for compact hypersurfaces in both Riemannian and spacelike settings and used it to characterize these hypersurfaces in terms of their scalar curvature. These works highlight the essential role of the conformality of vector fields in understanding the geometry of submanifolds. For more recent references in this field, see [12,13,14,15,16,17].

In this article, we investigate compact Riemannian hypersurfaces $(M,g)$ that are isometrically immersed in a Riemannian or Lorentzian manifold $(\overline{M},\overline{g})$ of dimension $n+1$ endowed with a CCVF $\overline{\zeta }$. In the case of Lorentzian manifolds, we consider $\overline{\zeta }$ as a timelike vector field, and we denote its restriction to M by $\zeta$. The support function $\vartheta$ of M is defined as the component of $\zeta$ along the unit-normal vector field N to M, which defines the smooth function $\vartheta =\overline{g}\left(\zeta ,N\right)$. This function plays a fundamental role in our analysis. This function has been used by many authors (see for example [11,12,18,19,20,21]). We derive an expression for its Laplacian $\Delta \vartheta$ (Theorem 1). With this formulation, we establish a generalized integral formula for such hypersurfaces (Theorem 2 and Theorem 3). Applying this formula, we obtain geometric characterizations of compact hypersurfaces, particularly when the ambient space $\overline{M}$ has constant sectional curvature $\overline{c}$. Our findings cover important cases such as hypersurfaces in Euclidean space ${\mathbb{R}}^{n+1}$, the sphere $S^{n+1}$, and de Sitter space $S_1^{n+1}$. In particular, we establish conditions under which hypersurfaces in these spaces are characterized as spheres, illustrating the significance of CCVFs in hypersurface geometry. This work extends and unifies the previous results in the field, such as the results in [6,9,11,12,20,21,22], by providing new tools for the study of compact Riemannian hypersurfaces in both Riemannian and Lorentzian manifolds.

This paper is organized as follows: Section 2 introduces the basic concepts and fundamental formulae in the theory of hypersurfaces in pseudo-Riemannian manifolds, along with a discussion on conformal vector fields and their properties. In Section 3, we study extrinsic spheres, defined as totally umbilical hypersurfaces with nonzero constant mean curvature in pseudo-spheres, and present small spheres in de Sitter space as a key example. Section 4 presents the derivation of the Laplacian of the support function and the integral formula for hypersurfaces admitting CCVFs. In Section 5, we derive integral formulae for compact Riemannian hypersurfaces immersed in Riemannian or Lorentzian manifolds with constant sectional curvatures that also possess CCVFs. These results highlight the central role of CCVFs in the study of hypersurfaces. Finally, in Section 6, we utilize these findings to obtain geometric characterizations of hypersurfaces in ambient spaces of constant sectional curvature, including Euclidean space ${\mathbb{R}}^{n+1}$, the sphere $S^{n+1}$, and de Sitter space $S_1^{n+1}.$

2. Preliminaries

Let $n\ge 2$, and consider a pseudo-Riemannian manifold $\left(\overline{M},\overline{g}\right)$ of dimension $n+1$, where the metric $\overline{g}$ is either Riemannian or Lorentzian, corresponding to signatures $(0,n+1)$ and $(1,n)$, respectively. Let $(M,g)$ denote an n-dimensional connected Riemannian manifold that is isometrically immersed in $\left(\overline{M},\overline{g}\right)$. In this context, $(M,g)$ will be considered a Riemannian hypersurface in $\left(\overline{M},\overline{g}\right)$, and, if $\left(\overline{M},\overline{g}\right)$ is Lorentzian, $(M,g)$ will be regarded as a spacelike hypersurface of $\left(\overline{M},\overline{g}\right)$.

In this work, we consistently use the sign and notation conventions as presented in [23,24]. According to these references, the Levi-Civita connections, $\overline{\nabla }$ of $\left(\overline{M},\overline{g}\right)$ and ∇ of $\left(M,g\right)$, are related by the Gauss formula, which, for any vector fields X, Y on M, is given by

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

The shape operator A of the hypersurface $\left(M,g\right)$ in $\left(\overline{M},\overline{g}\right)$, relative to the normal vector field N, is defined by the Weingarten formula (see [24])

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

The curvature tensor R of the hypersurface $\left(M,g\right)$ can be expressed in terms of the shape operator A and the curvature tensor $\overline{R}$ of $\left(\overline{M},\overline{g}\right)$ through the Gauss equation as follows

$$R\left(X,Y\right)Z={\left(\overline{R}\left(X,Y\right)Z\right)}^{\top }+ϵ\left[\overline{g}\left(A\left(Y\right),Z\right)A\left(X\right)-\overline{g}\left(A\left(X\right),Z\right)A\left(Y\right)\right],$$

for all vector fields $X,Y,Z\in \mathfrak{X}\left(M\right)$, where ${\left(\overline{R}\left(X,Y\right)Z\right)}^{\top }$ is the tangential component of $\overline{R}\left(X,Y\right)Z$. Consequently, the Ricci curvature $Ric$ of $\left(M,g\right)$ is given by

$$Ric\left(X,Y\right)=\overline{Ric}\left(X,Y\right)-ϵ\overline{g}\left(\overline{R}\left(N,X\right)Y,N\right)+g\left(\left(A\left(X\right),nHY-ϵA\left(Y\right)\right)\right),$$

(1)

for all $X,Y\in \mathfrak{X}\left(M\right)$, where $\overline{Ric}$ is the Ricci curvature of $\left(\overline{M},\overline{g}\right)$, and the mean curvature H of $\left(M,g\right)$ is given by

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

The Codazzi equation defines the normal component of $\overline{R}(X,Y)Z$ in relation to the derivative of the shape operator A, and it is expressed as

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

(2)

where $\nabla A$ is the covariant derivative of A (see [23,24]), which is

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

(3)

Recall that a vector field $\overline{\zeta }\in \mathfrak{X}\left(\overline{M}\right)$ is called conformal if its Lie derivative satisfies

$$L_{\overline{\zeta }}\overline{g}\left(X,Y\right)=2\psi \overline{g}\left(X,Y\right),$$

for all $X,Y\in \mathfrak{X}\left(\overline{M}\right)$, where $\psi$ is a smooth function, known as the potential function of $\overline{\zeta }$.

A conformal vector field $\overline{\zeta }$ is called homothetic if its potential function $\psi$ is constant and is a Killing vector field if $\psi =0$. Furthermore, $\overline{\zeta }$ is said to be closed conformal (or concircular) if its covariant derivative satisfies

$${\overline{\nabla }}_X\overline{\zeta }=\psi X,$$

for all $X\in \mathfrak{X}\left(\overline{M}\right)$. In particular, $\overline{\zeta }$ is called concurrent if $\psi$ is a nonzero constant and parallel if $\psi$ is identically zero [23].

Let $\overline{\zeta }\in \mathfrak{X}\left(\overline{M}\right)$ be a conformal vector field (and, in particular, a CCVF) on $\overline{M}$, which we assume to be timelike when $\left(\overline{M},\overline{g}\right)$ is Lorentzian. In this case, we can choose a globally defined unit timelike vector field N, normal to M and aligned with the time orientation of $\overline{\zeta }$. This ensures that $\overline{g}\left(\overline{\zeta },N\right)<0$ throughout M.

Let $\zeta$ represent the restriction of the conformal vector field $\overline{\zeta }$ to M. Consequently, the expression

$$\vartheta =\overline{g}\left(N,\zeta \right)$$

defines a smooth function on M, known as the support function of $\zeta$.

In the case of Lorentzian manifolds, the reverse Cauchy–Schwarz inequality yields

$$\left|\vartheta \right|=\left|\overline{g}\left(N,\zeta \right)\right|\ge \left|N\right|\left|\zeta \right|,$$

(see [24], Propsition 30, P.144). Since N and $\zeta$ have been chosen to be in the same time cone (i.e., $\vartheta =\overline{g}\left(N,\zeta \right)<0$), and $\left|N\right|=\sqrt{-\overline{g}\left(N,N\right)}=1$, we obtain

$$-\vartheta =-\overline{g}\left(N,\zeta \right)\ge \left|\zeta \right|=\sqrt{-\overline{g}\left(\zeta ,\zeta \right)},$$

which yields

$$\vartheta \le -\sqrt{-\overline{g}\left(\zeta ,\zeta \right)}.$$

Let ${\zeta }^{\top }$ be the tangential component of $\zeta$ to M. Then, we have the decomposition

$$\zeta ={\zeta }^{\top }+ϵ\vartheta N,$$

where $ϵ=\overline{g}\left(N,N\right)=\pm 1,$ depending on whether $\left(\overline{M},\overline{g}\right)$ is Riemannian or Lorentzian. Conformal vector fields of this type have been extensively studied in the literature. In particular, [25,26,27] explore various properties of such vector fields. Moreover, in [11,12], an expression for the Laplacian $\Delta \vartheta$ of the function $\vartheta =\overline{g}(N,\zeta )$ is derived, where $\zeta$ is a conformal vector field and N is a globally defined unit-normal vector field. This expression relates $\Delta \vartheta$ to the Ricci curvature $\overline{Ric}$ of $\left(\overline{M},\overline{g}\right)$ and the norm of the shape operator A of $\left(M,g\right)$, and it is given by

$$\Delta \vartheta =-ϵ\vartheta \left(\overline{Ric}\left(N,N\right)+{∥A∥}^2\right)-ϵn\left(\psi H+ϵ\overline{g}\left(\overline{\nabla }\psi ,N\right)+g\left(\nabla H,{\zeta }^{\top }\right)\right).$$

(4)

Additionally, a useful expression for $\mathrm{div}\left({\left({\overline{\nabla }}_N\zeta \right)}^{\top }\right)$ is provided in [11]

$$\mathrm{div}\left({\left({\overline{\nabla }}_N\zeta \right)}^{\top }\right)=\overline{Ric}\left(N,\zeta \right)+n\overline{g}\left(\overline{\nabla }\psi ,N\right).$$

Substituting $\zeta ={\zeta }^{\top }+ϵ\vartheta N$ into the above equation, we obtain

$$\mathrm{div}\left({\left({\overline{\nabla }}_N\zeta \right)}^{\top }\right)=\overline{Ric}\left(N,{\zeta }^{\top }\right)+ϵ\vartheta \overline{Ric}\left(N,N\right)+n\overline{g}\left(\overline{\nabla }\psi ,N\right).$$

(5)

Using this result, we derive the following integral formula for compact hypersurfaces in $\left(\overline{M},\overline{g}\right)$

$$\underset{M}{\int }\left(\overline{Ric}\left(N,{\zeta }^{\top }\right)+ϵ\vartheta \overline{Ric}\left(N,N\right)+n\overline{g}\left(\overline{\nabla }\psi ,N\right)\right)dV=0.$$

Remark 1. 

Note that $N\left(\psi \right)=\overline{g}\left(\overline{\nabla }\psi ,N\right)$ is not necessarily zero in general, as discussed in [11,19].

From now on, we will use $∥·∥$ to denote the norm of operators and $\left|·\right|$ to denote the norm of vector fields.

3. Extrinsic Spheres

Consider a pseudo-sphere $S_{\nu }^{n+1}\left(\overline{c}\right)$ of index $\nu$ (where $\nu =0$ or 1), with $\overline{c}>0$. For $\nu =0,$ $S^{n+1}\left(\overline{c}\right)$ represents the standard sphere in Euclidean space ${\mathbb{R}}^{n+2}$. When $\nu =1$, this corresponds to the de Sitter space $S_1^{n+1}\left(\overline{c}\right)$ in Minkowski space ${\mathbb{R}}_1^{n+2}$. Recall that a hypersurface $\left(M,g\right)$ of the pseudo-sphere $S_{\nu }^{n+1}\left(\overline{c}\right)$ is said to be totally umbilical if its shape operator A satisfies $A=\mathrm{\Phi }I$ for some scalar function $\mathrm{\Phi }$, where I denotes the identity operator. In particular, if $A\equiv 0$, then M is called totally geodesic (cf. [24]).

An extrinsic sphere is defined as a totally umbilical hypersurface with a nonzero constant mean curvature [24]. These hypersurfaces are of particular interest in differential geometry as they generalize the classical concept of spheres in Euclidean space to the general setting of pseudo-Riemannian manifolds. Small spheres in spaces of constant curvature, such as Euclidean spheres, hyperbolic spaces, and de Sitter spaces, are standard examples of extrinsic spheres (see [28]). In what follows, we present a specific example of a small sphere in de Sitter space.

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

$$\begin{array}{c}\hfill S^n\left(c\right)=\left\{\left(Z_1,\dots ,Z_{n+2}\right)\in {\mathbb{R}}_1^{n+2}:-Z_1^2+{\displaystyle \sum _{i=2}^{n+2}}{Z}_i^2={\displaystyle \frac{1}{\overline{c}}},Z_1=\sqrt{{\displaystyle \frac{1}{c}}-{\displaystyle \frac{1}{\overline{c}}}}\right\}.\end{array}$$

defines a spacelike hypersurface in $S_1^{n+1}\left(\overline{c}\right)$, with a timelike unit-normal vector field

$$\begin{array}{c}\hfill N_s\left(Z_1,\dots ,Z_{n+2}\right)=\left(-\sqrt{{\displaystyle \frac{\overline{c}}{c}}},\sqrt{\overline{c}-c}{Z}_2,\dots ,\sqrt{\overline{c}-c}{Z}_{n+2}\right),\end{array}$$

for all $\left(Z_1,\dots ,Z_{n+2}\right)\in S^n\left(c\right).$

The Levi-Civita connection satisfies

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

for all $X\in \mathfrak{X}\left(S^n\left(c\right)\right).$ The corresponding shape operator A satisfies

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

As $c<\overline{c}$, the mean curvature H of $S^n\left(c\right)$ is a nonzero constant. This implies that $S^n\left(c\right)$ is a totally umbilical submanifold of the de Sitter space $S_1^{n+1}\left(\overline{c}\right),$ which is not totally geodesic.

Examples of small spheres in Riemannian geometry have been discussed in [29].

4. Integral Formulae for Compact Riemannian Hypersurfaces in Spaces Admitting a CCVF

Let $\left(M,g\right)$ be an orientable n-dimensional Riemannian hypersurface immersed in an $\left(n+1\right)$-dimensional Riemannian or Lorentzian manifold $\left(\overline{M},\overline{g}\right)$. Let $\overline{\zeta }\in \mathfrak{X}\left(\overline{M}\right)$ be a CCVF, assumed to be timelike when $\left(\overline{M},\overline{g}\right)$ is Lorentzian, and let $\zeta$ denote its restriction to M. Before presenting our main result, we derive a useful expression for the Laplacian $\Delta \vartheta$ in the case where $\zeta$ is closed conformal.

For any $X\in \mathfrak{X}\left(M\right)$, using the fact that $\zeta$ is a CCVF, we apply the Gauss and Weingarten formulae to obtain

$${\overline{\nabla }}_X\zeta ={\nabla }_X{\zeta }^{\top }-ϵ\vartheta A\left(X\right)+ϵg\left(A\left({\zeta }^{\top }\right)+\nabla \vartheta ,X\right)N,$$

where ${\zeta }^{\top }$ is the tangential component of $\zeta$ to M, and N is a globally defined unit-normal vector field to M. By equating the tangential and normal components of the above equation, we deduce that

$${\nabla }_X{\zeta }^{\top }=\psi X+ϵ\vartheta A\left(X\right),$$

(6)

and

$$\nabla \vartheta =-A\left({\zeta }^{\top }\right).$$

(7)

Here, $\nabla \vartheta$ represents the gradient of $\vartheta$. We deduce from (6) and the definition of divergence that

$$\mathrm{div}\left({\zeta }^{\top }\right)=n\left(\psi +\vartheta H\right).$$

(8)

We now present a useful expression for the Laplacian $\Delta \vartheta$ of the support function $\vartheta$.

Theorem 1. 

Let $\left(\overline{M},\overline{g}\right)$ be an $\left(n+1\right)$-dimensional pseudo-Riemannian manifold, either Riemannian or Lorentzian, equipped with a CCFV $\overline{\zeta }\in \mathfrak{X}\left(\overline{M}\right)$. If $\left(\overline{M},\overline{g}\right)$ is Lorentzian, assume that $\overline{\zeta }$ is timelike. Let $\left(M,g\right)$ be a connected Riemannian hypersurface of $\left(\overline{M},\overline{g}\right)$. Let $\zeta ,{\zeta }^{\top },\vartheta$, and N be the same as above. Then, the Laplacian $\Delta \vartheta$ of the function $\vartheta =\overline{g}\left(N,\zeta \right)$ is given by

$$\Delta \vartheta =\overline{Ric}\left({\zeta }^{\top },N\right)-ϵ\vartheta {∥A∥}^2-ϵn\left(\psi H+g\left(\nabla H,{\zeta }^{\top }\right)\right).$$

(9)

Proof. 

Consider a local orthonormal frame $\left\{e_1,e_2,...,e_n\right\}$ on M. By applying the divergence operator to both sides of (7), we obtain

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

Using (3), we obtain

$$\Delta \vartheta =-{\displaystyle \sum _{i=1}^n}g\left(\left({\nabla }_{e_i}A\right)\left({\zeta }^{\top }\right),e_i\right)-{\displaystyle \sum _{i=1}^n}g\left(A\left({\nabla }_{e_i}{\zeta }^{\top }\right),e_i\right).$$

From (6), it follows that

$$\Delta \vartheta =-{\displaystyle \sum _{i=1}^n}g\left(\left({\nabla }_{e_i}A\right)\left({\zeta }^{\top }\right),e_i\right)-\psi \mathrm{trace}\left(A\right)-ϵ\vartheta {∥A∥}^2.$$

Since A is symmetric, we have

$$\Delta \vartheta =-{\displaystyle \sum _{i=1}^n}g\left(\left({\nabla }_{e_i}A\right)e_i,{\zeta }^{\top }\right)-\psi \mathrm{trace}\left(A\right)-ϵ\vartheta {∥A∥}^2.$$

(10)

To complete the proof, we compute

$$\begin{array}{c}\hfill ϵng\left(\nabla H,{\zeta }^{\top }\right)={\displaystyle \sum _{i=1}^n}g\left(\left(\nabla A\right)\left({\zeta }^{\top },e_i\right),e_i\right).\end{array}$$

Using (2), we find

$$\begin{array}{ccc}\hfill ϵng\left(\nabla H,{\zeta }^{\top }\right)& =& {\displaystyle \sum _{i=1}^n}\overline{g}\left(\overline{R}\left({\zeta }^{\top },e_i\right)e_i,N\right)+{\displaystyle \sum _{i=1}^n}\overline{g}\left(\left({\nabla }_{e_i}A\right)\left({\zeta }^{\top }\right),e_i\right)\hfill \\ & =& \overline{Ric}\left({\zeta }^{\top },N\right)+{\displaystyle \sum _{i=1}^n}\overline{g}\left(\left({\nabla }_{e_i}A\right)\left({\zeta }^{\top }\right),e_i\right).\hfill \end{array}$$

By the symmetry of A, we have

$$ϵng\left(\nabla H,{\zeta }^{\top }\right)=\overline{Ric}\left({\zeta }^{\top },N\right)+{\displaystyle \sum _{i=1}^n}\overline{g}\left(\left({\nabla }_{e_i}A\right)e_i,{\zeta }^{\top }\right).$$

Therefore,

$${\displaystyle \sum _{i=1}^n}\overline{g}\left(\left({\nabla }_{e_i}A\right)e_i,{\zeta }^{\top }\right)=-\overline{Ric}\left({\zeta }^{\top },N\right)+ϵng\left(\nabla H,{\zeta }^{\top }\right).$$

Substituting into (10), we obtain the desired formula. □

Remark 2. 

Formula (9) in Theorem 1 can also be derived through an alternative method. By using (5), and considering ζ to be closed conformal, ${\overline{\nabla }}_N\zeta =\psi N$ for some smooth function ψ, which implies that

$${\left({\overline{\nabla }}_N\zeta \right)}^{\top }=0.$$

Consequently, it follows that

$$\overline{Ric}\left(N,{\zeta }^{\top }\right)=-ϵ\vartheta \overline{Ric}\left(N,N\right)-n\overline{g}\left(\overline{\nabla }\psi ,N\right).$$

Substituting this expression into Equation (4) yields (9).

With the help of (9), we derive an integral formula for a compact Riemannian hypersurface in $\left(\overline{M},\overline{g}\right)$.

Theorem 2. 

Let $\left(\overline{M},\overline{g}\right)$ be a pseudo-Riemannian manifold of dimension $\left(n+1\right)$, which can be either Riemannian or Lorentzian, equipped with a CCFV $\overline{\zeta }\in \mathfrak{X}\left(\overline{M}\right)$. If $\left(\overline{M},\overline{g}\right)$ is Lorentzian, assume that $\overline{\zeta }$ is timelike. Let $\left(M,g\right)$ be a compact Riemannian hypersurface of $\left(\overline{M},\overline{g}\right)$. Let $\zeta ,{\zeta }^{\top },\vartheta$, and N be the same as above. Then, the following integral formula holds

$$\begin{array}{ccc}\hfill \underset{M}{\int }{\vartheta }^2\left({∥A∥}^2-n{H}^2\right)dV& =& \underset{M}{\int }\left(n\left(n-1\right)\left({\vartheta }^2{H}^2-{\psi }^2\right)-Ric\left({\zeta }^{\top },{\zeta }^{\top }\right)+\overline{Ric}\left(\zeta ,{\zeta }^{\top }\right)\right)dV\hfill \\ & & -\underset{M}{\int }\left(g\left(\nabla \psi ,{\zeta }^{\top }\right)+ϵ\overline{g}\left(\overline{R}\left(N,{\zeta }^{\top }\right){\zeta }^{\top },N\right)\right)dV\hfill \end{array}$$

(11)

Proof. 

From (9), we obtain

$$\vartheta \Delta \vartheta =\vartheta \overline{Ric}\left({\zeta }^{\top },N\right)-ϵ{\vartheta }^2{∥A∥}^2-ϵn\vartheta \left(\psi H+g\left(\nabla H,{\zeta }^{\top }\right)\right).$$

Equivalently,

$$\mathrm{div}\left(\vartheta \Delta \vartheta \right)-{\left|\nabla \vartheta \right|}^2=\vartheta \overline{Ric}\left({\zeta }^{\top },N\right)-ϵ{\vartheta }^2{∥A∥}^2-ϵn\vartheta \left(\psi H+g\left(\nabla H,{\zeta }^{\top }\right)\right).$$

By applying the divergence theorem and properties of shape operator A, we obtain

$$\begin{array}{ccc}\hfill \mathrm{div}\left(H\left(\vartheta {\zeta }^{\top }\right)\right)& =& \vartheta g\left(\nabla H,{\zeta }^{\top }\right)+H\mathrm{div}\left(\vartheta {\zeta }^{\top }\right)\hfill \\ & =& \vartheta g\left(\nabla H,{\zeta }^{\top }\right)+H\left(\vartheta \mathrm{div}\left({\zeta }^{\top }\right)+Hg\left(\nabla \vartheta ,{\zeta }^{\top }\right)\right)\hfill \\ & =& \vartheta g\left(\nabla H,{\zeta }^{\top }\right)+n\vartheta H\left(\psi +\vartheta H\right)-Hg\left(A\left({\zeta }^{\top }\right),{\zeta }^{\top }\right),\hfill \end{array}$$

which yields

$$\vartheta g\left(\nabla H,{\zeta }^{\top }\right)=\mathrm{div}\left(H\left(\vartheta {\zeta }^{\top }\right)\right)-n\vartheta H\left(\psi +\vartheta H\right)+Hg\left(A\left({\zeta }^{\top }\right),{\zeta }^{\top }\right).$$

It follows that

$$\begin{array}{ccc}\hfill \mathrm{div}\left(\vartheta \Delta \vartheta \right)-{\left|\nabla \vartheta \right|}^2& =& \vartheta \overline{Ric}\left({\zeta }^{\top },N\right)-ϵ{\vartheta }^2{∥A∥}^2-ϵn\vartheta \left(\psi H\right)-ϵn(\mathrm{div}\left(H\left(\vartheta {\zeta }^{\top }\right)\right)\hfill \\ & & -n\vartheta H\left(\psi +\vartheta H\right)+Hg\left(A\left({\zeta }^{\top }\right),{\zeta }^{\top }\right)).\hfill \end{array}$$

Equivalently,

$$\begin{array}{ccc}\hfill \mathrm{div}\left(\vartheta \Delta \vartheta \right)-{\left|\nabla \vartheta \right|}^2+ϵn\mathrm{div}\left(H\left(\vartheta {\zeta }^{\top }\right)\right)& =& \vartheta \overline{Ric}\left({\zeta }^{\top },N\right)-ϵ{\vartheta }^2{∥A∥}^2+ϵn\left(n-1\right)\vartheta \psi H\hfill \\ & & +ϵn^2{\vartheta }^2{H}^2-ϵnHg\left(A\left({\zeta }^{\top }\right),{\zeta }^{\top }\right).\hfill \end{array}$$

(12)

From (1), we obtain

$$\begin{array}{ccc}\hfill -{\left|\nabla \vartheta \right|}^2& =& ϵRic\left({\zeta }^{\top },{\zeta }^{\top }\right)-ϵ\overline{Ric}\left({\zeta }^{\top },{\zeta }^{\top }\right)+\overline{g}\left(\overline{R}\left(N,{\zeta }^{\top }\right){\zeta }^{\top },N\right)\hfill \\ & & -ϵnHg\left(A\left({\zeta }^{\top }\right),{\zeta }^{\top }\right).\hfill \end{array}$$

(13)

Since

$$\mathrm{div}\left(\psi {\zeta }^{\top }\right)=n{\psi }^2+n\vartheta \psi H+g\left(\nabla \psi ,{\zeta }^{\top }\right),$$

and

$$\begin{array}{c}\hfill \overline{Ric}\left({\zeta }^{\top },{\zeta }^{\top }\right)+ϵ\vartheta \overline{Ric}\left(N,{\zeta }^{\top }\right)=\overline{Ric}\left(\zeta ,{\zeta }^{\top }\right),\end{array}$$

it follows from (12) and (13)

$$\begin{array}{ccc}& & Ric\left({\zeta }^{\top },{\zeta }^{\top }\right)+ϵ\overline{g}\left(\overline{R}\left(N,{\zeta }^{\top }\right){\zeta }^{\top },N\right)+ϵ\mathrm{div}\left(\vartheta \Delta \vartheta \right)+n\mathrm{div}\left(H\left(\vartheta {\zeta }^{\top }\right)\right)\hfill \\ & =& \overline{Ric}\left(\zeta ,{\zeta }^{\top }\right)-{\vartheta }^2\left({∥A∥}^2-n{H}^2\right)+\left(n-1\right)\mathrm{div}\left(\psi {\zeta }^{\top }\right)-\left(n-1\right)g\left(\nabla \psi ,{\zeta }^{\top }\right)\hfill \\ & & +n\left(n-1\right)\left({\vartheta }^2{H}^2-{\psi }^2\right).\hfill \end{array}$$

Integrating both sides of the equation above and noting that M is compact, formula (11) is obtained. □

The integral formula (11) in Theorem 2 can alternatively be expressed in terms of the square of div $\left({\zeta }^{\top }\right)$ as follows.

Theorem 3. 

Under the same assumptions as Theorem 2, we obtain

$$\begin{array}{ccc}\hfill \underset{M}{\int }{\vartheta }^2\left({∥A∥}^2-n{H}^2\right)dV& =& \underset{M}{\int }\left({\displaystyle \frac{n-1}{n}}{\left(\mathit{div}\left({\zeta }^{\top }\right)\right)}^2-Ric\left({\zeta }^{\top },{\zeta }^{\top }\right)+\overline{Ric}\left(\zeta ,{\zeta }^{\top }\right)\right)dV\hfill \\ & & +\underset{M}{\int }\left(\left(n-1\right)g\left(\nabla \psi ,{\zeta }^{\top }\right)-ϵ\overline{g}\left(\overline{R}\left(N,{\zeta }^{\top }\right){\zeta }^{\top },N\right)\right)dV.\hfill \end{array}$$

(14)

Proof. 

Using (8), we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{n-1}{n}}{\left(\mathrm{div}\left({\zeta }^{\top }\right)\right)}^2& =& n\left(n-1\right)\left({\psi }^2+{\vartheta }^2{H}^2+2\psi \vartheta H\right)\hfill \\ & =& n\left(n-1\right)\left({\vartheta }^2{H}^2-{\psi }^2+2{\psi }^2+2\psi \vartheta H\right)\hfill \\ & =& n\left(n-1\right)\left({\vartheta }^2{H}^2-{\psi }^2+2\psi \left(\psi +\vartheta H\right)\right)\hfill \\ & =& n\left(n-1\right)\left({\vartheta }^2{H}^2-{\psi }^2\right)+2\left(n-1\right)\psi \mathrm{div}\left({\zeta }^{\top }\right)\hfill \end{array}$$

Recalling that

$$\begin{array}{c}\hfill \psi \mathrm{div}\left({\zeta }^{\top }\right)=\mathrm{div}\left(\psi {\zeta }^{\top }\right)-g\left(\nabla \psi ,{\zeta }^{\top }\right),\end{array}$$

we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{n-1}{n}}{\left(\mathrm{div}\left({\zeta }^{\top }\right)\right)}^2& =& n\left(n-1\right)\left({\vartheta }^2{H}^2-{\psi }^2\right)\hfill \\ & & +2\left(n-1\right)\left(\mathrm{div}\left(\psi {\zeta }^{\top }\right)-g\left(\nabla \psi ,{\zeta }^{\top }\right)\right),\hfill \end{array}$$

which yields

$$\begin{array}{ccc}\hfill n\left(n-1\right)\left({\vartheta }^2{H}^2-{\psi }^2\right)& =& {\displaystyle \frac{n-1}{n}}{\left(\mathrm{div}\left({\zeta }^{\top }\right)\right)}^2-2\left(n-1\right)\left(\mathrm{div}\left(\psi {\zeta }^{\top }\right)-g\left(\nabla \psi ,{\zeta }^{\top }\right)\right)\hfill \end{array}$$

By integrating both sides of the above equation, we obtain formula (14). □

5. Integral Formulae for Compact Riemannian Hypersurfaces in Spaces of Constant Sectional Curvature

In this section, we examine the previous results under the assumption that the ambient space $\left(\overline{M},\overline{g}\right)$ has constant sectional curvature $\overline{c}$. In this setting, the curvature tensor $\overline{R}$ of $\overline{M}$ satisfies

$$\overline{R}\left(N,X\right)N=-ϵ\overline{c}X,$$

and the Ricci curvature of $\overline{M}$ is given by

$$\overline{Ric}\left(X,Y\right)=n\overline{c}\overline{g}\left(X,Y\right),$$

for all vector fields $X,Y\in \mathfrak{X}\left(\overline{M}\right)$, where N is a unit-normal vector field, $ϵ=\overline{g}\left(N,N\right)=\pm 1,$ depending on whether $\left(\overline{M},\overline{g}\right)$ is Riemannian or Lorentzian, and n denotes the dimension of the hypersurface $\left(M,g\right)$.

As a consequence of Theorem 2, when applied to $\left(\overline{M}\left(\overline{c}\right),\overline{g}\right)$ with constant sectional curvature $\overline{c}$, we obtain the following result. It is worth noting that the only spaces of constant curvature that admit small spheres $S^n$ of dimension n are the Euclidean space ${\mathbb{R}}^{n+1}$, the sphere $S^{n+1}$, hyperbolic space ${\mathbb{H}}^{n+1}$, and de Sitter space $S_1^{n+1}$ (see [28]).

Theorem 4. 

Let $\left(\overline{M}\left(\overline{c}\right),\overline{g}\right)$ be an $\left(n+1\right)$-dimensional Riemannian or Lorentizan manifold with constant sectional curvature $\overline{c}$, and let $\left(M,g\right)$ be a compact Riemannian hypersurface immersed in $\left(\overline{M}\left(\overline{c}\right),\overline{g}\right)$. Suppose $\overline{\zeta }\in \mathfrak{X}\left(\overline{M}\right)$ is a CCVF, which is assumed to be timelike when $\left(\overline{M}\left(\overline{c}\right),\overline{g}\right)$ is Lorentzian. Let ζ denote its restriction to M, and let N be a globally defined unit-normal vector field along M. Define ${\zeta }^{\top }$ as the tangential component of ζ, and let $\vartheta =\overline{g}\left(\zeta ,N\right)$ be the support function. Then, the following integral formula holds:

$$\begin{array}{ccc}\hfill \underset{M}{\int }{\vartheta }^2\left({∥A∥}^2-n{H}^2\right)dV& =& \underset{M}{\int }\left(n\left(n-1\right)\left({\vartheta }^2{H}^2-{\psi }^2\right)-Ric\left({\zeta }^{\top },{\zeta }^{\top }\right)\right)dV\hfill \\ & +& \left(n-1\right)\underset{M}{\int }\left(\overline{c}{\left|{\zeta }^{\top }\right|}^2-g\left(\nabla \psi ,{\zeta }^{\top }\right)\right)dV.\hfill \end{array}$$

(15)

Proof. 

Since $\overline{M}\left(\overline{c}\right)$ has constant sectional curvature $\overline{c}$, it follows that

$$\overline{Ric}\left({\zeta }^{\top },{\zeta }^{\top }\right)=n\overline{c}{\left|{\zeta }^{\top }\right|}^2$$

(16)

and

$$\overline{g}\left(\overline{R}\left(N,{\zeta }^{\top }\right){\zeta }^{\top },N\right)=ϵ\overline{c}{\left|{\zeta }^{\top }\right|}^{2.}$$

(17)

Substituting these expressions into (11) produces the desired formula. □

Using Formula (14) in Theorem 3 and considering $\overline{M}\left(\overline{c}\right)$ with constant sectional curvature $\overline{c}$, we deduce the following result.

Theorem 5. 

Let the notation and assumptions be as in Theorem 4. Then, we have

$$\begin{array}{ccc}\hfill \underset{M}{\int }{\vartheta }^2\left({∥A∥}^2-n{H}^2\right)dV& =& \underset{M}{\int }\left({\displaystyle \frac{n-1}{n}}{\left(\mathit{div}\left({\zeta }^{\top }\right)\right)}^2-Ric\left({\zeta }^{\top },{\zeta }^{\top }\right)\right)dV\hfill \\ & +& \left(n-1\right)\underset{M}{\int }\left(\overline{c}{\left|{\zeta }^{\top }\right|}^2+g\left(\nabla \psi ,{\zeta }^{\top }\right)\right)dV.\hfill \end{array}$$

(18)

Proof. 

The proof follows in a similar manner by substituting (16) and (17) into (14). □

As a consequence of the previous theorems, we derive integral formulae for the special case where $\overline{\zeta }$ is a concurrent vector field on $\left(\overline{M},\overline{g}\right)$, that is, a vector field satisfying

$${\overline{\nabla }}_X\overline{\zeta }=CX,$$

for all $X\in \mathfrak{X}\left(\overline{M}\right)$, where C is a nonzero constant.

Corollary 1. 

Let the notation and assumptions be as stated in Theorem 4, and assume, in addition, that $\overline{\zeta }$ is a concurrent vector field with a nonzero constant factor C. In this case, the following integral formula holds:

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

In particular, $\left(M,g\right)$ is an extrinsic sphere if and only if

$$\underset{M}{\int }Ric\left({\zeta }^{\top },{\zeta }^{\top }\right)dV\ge \underset{M}{\int }\left(n\left(n-1\right){\vartheta }^2{H}^2+\overline{c}{\left|{\zeta }^{\top }\right|}^2\right)dV,$$

Corollary 2. 

Under the above notation and assumptions, it follows that

$$\begin{array}{ccc}\hfill \underset{M}{\int }{\vartheta }^2\left({∥A∥}^2-n{H}^2\right)dV& =& \underset{M}{\int }\left({\displaystyle \frac{n-1}{n}}{\left(\mathit{div}\left({\zeta }^{\top }\right)\right)}^2-Ric\left({\zeta }^{\top },{\zeta }^{\top }\right)\right)dV\hfill \\ & & +\left(n-1\right)\overline{c}\underset{M}{\int }{\left|{\zeta }^{\top }\right|}^{2}dV.\hfill \end{array}$$

In particular, if $\overline{c}\le 0$ and

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

then $\left(M,g\right)$ is an extrinsic sphere.

6. Applications

In this section, we present some consequences and applications of Theorems 4 and 5. Specifically, we apply the integral formulae (15) and (18) to hypersurfaces immersed in spaces of constant sectional curvature. As a first application, we focus on the case where the ambient space $\overline{M}\left(\overline{c}\right)$ is a pseudo-sphere $S_{\nu }^{n+1}\left(\overline{c}\right)$ of index $\nu$ (where $\nu =0$ or 1), with $\overline{c}>0$.

To establish our result, we first note that an important class of CCVFs on $S_v^{n+1}\left(\overline{c}\right)$ consists of those that are induced by a nonzero constant vector field Z on the pseudo-Euclidean space ${\mathbb{R}}_{\nu }^{n+2}$ of index $\nu$ (where $\nu =0$ or 1), ([21,22,30]). More precisely, let Z be a nonzero constant vector field on the pseudo-Euclidean space $\left({\mathbb{R}}_{\nu }^{n+2},〈·,·〉\right)$, which is assumed to be timelike when $\nu =1$. The tangential component of Z along the $S_{\nu }^{n+1}\left(\overline{c}\right)$, denoted by $\zeta =Z^{\top }$, satisfies

$$Z=\zeta +\overline{\sigma }\overline{N},$$

where $\overline{N}$ is the spacelike unit-normal vector field to $S_v^{n+1}\left(\overline{c}\right)$, and $\overline{\sigma }=〈Z,\overline{N}〉$ represents the support function. This leads to the relations

$${\nabla }_X\zeta =-\sqrt{\overline{c}}\overline{\sigma }X,\overline{\nabla }\overline{\sigma }=\sqrt{\overline{c}}\zeta ,$$

(19)

for all $X\in \mathfrak{X}\left(S_{\nu }^{n+1}\left(\overline{c}\right)\right)$. This implies that $\zeta$ is a nonzero CCVF on $S_{\nu }^{n+1}\left(\overline{c}\right)$ with potential function

$$\overline{\psi }=-\sqrt{\overline{c}}\overline{\sigma }.$$

We denote by $\sigma$ and $\psi$ the restrictions of the support function $\overline{\sigma }$ and the potential function $\overline{\psi }$ to the hypersurface M of $S_{\nu }^{n+1}\left(\overline{c}\right)$, respectively. As noted in Section 2, the vector field $\zeta$ can be expressed as

$$\zeta ={\zeta }^{\top }+\epsilon \vartheta N,$$

where ${\zeta }^{\top }$ represents the tangential component of the CCVF $\zeta$. From this, we deduce

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

(20)

By applying (20), we obtain

$${\zeta }^{\top }\left(\psi \right)=g\left(\nabla \psi ,{\zeta }^{\top }\right)=-\overline{c}{\left|{\zeta }^{\top }\right|}^2.$$

Substituting this into (15), we obtain a very useful formula.

Theorem 6. 

Let $\left(M,g\right)$ be a compact Riemannian hypersurface of the pseudo-sphere $\left(S_{\nu }^{n+1}\left(\overline{c}\right),\overline{g}\right)$, where $\nu =0$ or $1$ and $n\ge 2$. Let $\overline{\zeta }\in \mathfrak{X}\left(S_{\nu }^{n+1}\left(\overline{c}\right)\right)$ be a CCVF, assumed to be timelike when $\nu =1$. Let ζ be the restriction of $\overline{\zeta }$ to M, and let N be a globally defined unit-normal vector field to M. Define ${\zeta }^{\top }$ as the tangential component of ζ, and let $\vartheta =\overline{g}\left(\zeta ,N\right)$ be the support function. Then, the following integral formula holds:

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

Furthermore, using Theorem 5, this formula can be reformulated as

$$\underset{M}{\int }{\vartheta }^2\left({∥A∥}^2-n{H}^2\right)dV=\underset{M}{\int }\left({\displaystyle \frac{n-1}{n}}{\left(\mathit{div}\left({\zeta }^{\top }\right)\right)}^2-Ric\left({\zeta }^{\top },{\zeta }^{\top }\right)\right)dV.$$

As a nice consequence of the above results, we can derive several important properties and characterizations of totally umbilical hypersurfaces in the pseudo-sphere $S_{\nu }^{n+1}\left(\overline{c}\right)$, where $\nu =0$ or 1, as outlined in [21,22].

Theorem 7. 

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

$$\underset{M}{\int }Ric\left({\zeta }^{\top },{\zeta }^{\top }\right)dV\ge \left(n-1\right)\underset{M}{\int }\left(n\left({\vartheta }^2{H}^2-{\psi }^2\right)+2\overline{c}{\left|{\zeta }^{\top }\right|}^2\right)dV$$

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

Theorem 8. 

Under the same assumptions as above,

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

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

Next, we apply the integral formulae (15) and (18) to hypersurfaces in Euclidean space ${\mathbb{R}}^{n+1}$, where $\overline{c}=0$. These applications lead to characterizations of spheres in ${\mathbb{R}}^{n+1}$ based on the Ricci curvature and the divergence of the tangential component of the CCVF. The classification of such conformal vector fields has been extensively discussed in [26].

Theorem 9. 

Let $\left(M,g\right)$ be a compact Riemannian hypersurface immersed in $\left({\mathbb{R}}^{n+1},\overline{g}\right)$. Let $\overline{\zeta }\in {\mathbb{R}}^{n+1}$ be a CCVF. Let ζ be the restriction of $\overline{\zeta }$ to M, and let N be a globally defined unit-normal vector field to M. Define ${\zeta }^{\top }$ as the tangential component of ζ, and let $\vartheta =\overline{g}\left(\zeta ,N\right)$ be the support function. Then,

$${\vartheta }^2\left({∥A∥}^2-n{H}^2\right)=n\left(n-1\right)\left({\vartheta }^2{H}^2-{\psi }^2\right)-Ric\left({\zeta }^{\top },{\zeta }^{\top }\right),$$

or, equivalently,

$${\vartheta }^2\left({∥A∥}^2-n{H}^2\right)={\displaystyle \frac{n-1}{n}}{\left(\mathit{div}\left({\zeta }^{\top }\right)\right)}^2-Ric\left({\zeta }^{\top },{\zeta }^{\top }\right).$$

As a consequence of the above result, we obtain a characterization of spheres in ${\mathbb{R}}^{n+1}$ (see [20]).

Theorem 10. 

Under the same assumptions as in Theorem 9, the Ricci curvature of $\left(M,g\right)$ and the tangential component ${\zeta }^{\top }$ of ζ satisfy

$$Ric\left({\zeta }^{\top },{\zeta }^{\top }\right)\ge n\left(n-1\right)\left({\vartheta }^2{H}^2-{\psi }^2\right)$$

if and only if $\left(M,g\right)$ is isometric to a sphere. Equivalently, this condition can be expressed as

$$Ric\left({\zeta }^{\top },{\zeta }^{\top }\right)\ge {\displaystyle \frac{\left(n-1\right)}{n}}{\left(\mathit{div}\left({\zeta }^{\top }\right)\right)}^2$$

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

It is important to note that spheres do not exist in Minkowski or anti-de Sitter spaces (see [28]). However, extrinsic spheres in Lorentzian manifolds play a significant role in mathematical physics and relativity, particularly in the study of spacetime geometry.

A concrete example of such an extrinsic sphere is found in anti-de Sitter space $H_1^{n+1}\left(\overline{c}\right)$, where the only totally umbilical spacelike hypersurface is the hyperbolic space

$$H^n\left(c\right)=\left\{x\in H_1^{n+1}\left(\overline{c}\right):x_1=\sqrt{{\displaystyle \frac{1}{c}}-{\displaystyle \frac{1}{\overline{c}}}}\right\},$$

which has a shape operator $A=\pm \sqrt{\overline{c}-c}I$ and a constant sectional curvature c, satisfying the condition $c\le \overline{c}<0.$

7. Conclusions

This study has focused on compact Riemannian hypersurfaces immersed in ambient manifolds, both Riemannian and Lorentzian, that admit CCVFs. These vector fields play a crucial role in deriving integral formulae that relate the intrinsic and extrinsic geometry of hypersurfaces. We derived an integral formula involving the Laplacian of the support function associated with the projection of a CCVF onto the normal direction of the hypersurface. Using this formula, we developed new characterizations for compact hypersurfaces in spaces of constant sectional curvature. In particular, we established conditions under which such hypersurfaces must be totally umbilical. These findings generalize classical results in both Riemannian and Lorentzian settings, offering new directions for further research in pseudo-Riemannian geometry.

The assumption that the ambient space has constant sectional curvature is motivated by both geometric and analytical considerations in the study of integral formulae for compact Riemannian hypersurfaces. In such spaces, the curvature tensor has a simplified form, which makes it easier to compute and derive explicit integral expressions. These spaces also admit CCVFs, such as the position vector in Euclidean space and the tangential part of constant vector fields on the sphere. This makes such spaces ideal for examining the geometric role of CCVFs in hypersurface theory. Furthermore, many classical results in the study of hypersurfaces, such as [1,2,3], were initially formulated in Euclidean space. As a result, the constant curvature condition enables the generalization of these results to a wider range of ambient spaces.