Ricci Solitons on Riemannian Hypersurfaces Generated by Torse-Forming Vector Fields in Riemannian and Lorentzian Manifolds

Abstract

In this paper, we examine torse-forming vector fields to characterize extrinsic spheres (that is, totally umbilical hypersurfaces with nonzero constant mean curvatures) in Riemannian and Lorentzian manifolds. First, we analyze the properties of these vector fields on Riemannian manifolds. Next, we focus on Ricci solitons on Riemannian hypersurfaces induced by torse-forming vector fields of Riemannian or Lorentzian manifolds. Specifically, we show that such a hypersurface in the manifold with constant sectional curvature is either totally geodesic or an extrinsic sphere.

1. Introduction

Torse-forming vector fields require special attention due to their applications in relativity, cosmology, and the theory of submanifolds. These vector fields are important in various domains of physics and differential geometry (see [1,2,3]). Initially explored in [4] by K. Yano, torse-forming vector fields were defined and analyzed.

A vector field $\zeta$ on a pseudo-Riemannian manifold $(N,g)$ is called a torse-forming vector field if it satisfies the following equation:

$$D_X\zeta =\varphi X+\mu \left(X\right)\zeta ,$$

(1)

for any vector field X on $(N,g)$, where $\varphi$ is a smooth function on N, $\mu$ is a one-form, and D denotes the Levi–Civita connection of $(N,g)$. The one-form $\mu$ is called the generating form while the function $\varphi$ is called the conformal scalar [5]. When $\mu \left(\zeta \right)=0$, the vector field $\zeta$ is said to be a torqued vector field [6]. Let $\eta$ be the dual vector field corresponding to the generating form $\mu$, satisfying $\mu \left(X\right)=g(X,\eta )$ for every vector field X on $(N,g)$. As $\mu \left(\zeta \right)=g(\zeta ,\eta )$, then saying that $\zeta$ is a torqued vector field is equivalent to saying that $\zeta$ and $\eta$ are perpendicular. If the one-form $\mu$ is identically zero, then $\zeta$ is commonly known as a closed conformal (sometimes called concircular) vector field. When $\varphi$ is a non-zero constant and $\mu =0$, $\zeta$ is known as a concurrent vector field, and if $\varphi =0$ with $\mu$ is a non-zero constant, $\zeta$ is described as a recurrent vector field.

B.Y. Chen has made significant advancements in the theory of submanifolds [7,8,9]. Research related to submanifold theory and solitons in various geometric flows is being actively developed. It is also important to mention that Hamilton [10,11] introduced the concept of Ricci flow theory.

The study of Ricci solitons on hypersurfaces has gained attention refarding understanding the conditions that allow these structures to occur within Riemannian manifolds (see [12,13,14]). Although there has been extensive research on Ricci solitons in Riemannian manifolds, there is a notable lack of studies in Lorentzian manifolds. This gap highlights the critical importance of the study into Ricci solitons on Riemannian hypersurfaces within Lorentzian manifolds, as these concepts are integral to advancements in both geometry and physics.

This paper examines the properties of torse-forming vector fields on Riemannian manifolds. Subsequently, we will investigate the conditions under which a Riemannian hypersurface can evolve into a Ricci soliton within a Riemannian or Lorentzian manifold. This hypersurface is distinguished by the existence of a torse-forming vector field, which is essential in determining its geometric properties. Studies have indicated that Ricci solitons can occur on hypersurfaces within pseudo-Riemannian manifolds, as detailed in [15,16,17,18,19,20].

This study is organized as follows: Section 2 discusses the foundational concepts, detailing the basic principles and primary formulas in Ricci soliton theory. It also examines the fundamental concepts and key equations related to Riemannian hypersurfaces in both Riemannian and Lorentzian manifolds. In Section 3, we explore several findings on pseudo-Riemannian manifolds that feature a torse-forming vector field, accompanied by various useful formulas, from which important results are derived. Section 4 determines the critical conditions for Ricci solitons on a Riemannian hypersurface within either a Riemannian or Lorentzian manifold with a torse-forming vector field. We also examine the hypersurface in the manifold that has a constant sectional curvature, and we demonstrate that, under these conditions, it is totally umbilical; it possesses constant mean and sectional curvatures.

2. Preliminaries

For a pseudo-Riemannian manifold $(N,g)$ of dimension n, $n\ge 3$. Let $\mathfrak{X}\left(N\right)$ represent the set of all vector fields on N. Consider a tangent vector $X_p$ at a point p in N. A tangent vector $X_p$ at a point p is classified based on the evaluation of the metric tensor $g_p$ as follows: it is termed space-like if $g_p(X_p,X_p)>0$ or if $X_p=0$; it is considered time-like if $g_p(X_p,X_p)<0$; and it is identified as light-like if $g_p(X_p,X_p)=0$ with $X_p\ne 0$. Extending this classification, a vector field X on a manifold N is characterized as space-like, time-like, or light-like if it satisfies these criteria at each point within the manifold.

The Riemannian curvature tensor on N is a tensor field [21], which is expressed as follows:

$$R(X,Y)Z=D_{[X,Y]}Z-D_X{D}_{Y}Z+D_Y{D}_{X}Z,$$

(2)

for all $X,Y,Z\in \mathfrak{X}\left(N\right)$.

The Ricci curvature $Ric$ of $(N,g)$ is given by the following:

$$Ric(X,Y)={\displaystyle \sum _{i=1}^n}{\epsilon }_{i}g\left(R(X,E_i)Y,E_i\right),$$

(3)

for all $X,Y\in \mathfrak{X}\left(N\right)$, where $\{E_1,...,E_n\}$ is a local orthonormal frame on N, and ${\epsilon }_i=g(E_i,E_i)=\pm 1$.

The scalar curvature Scal of N is defined as follows:

$$\mathrm{Scal}={\displaystyle \sum _{i=1}^n}{\epsilon }_{i}Ric(E_i,E_i).$$

(4)

For $X\in \mathfrak{X}\left(N\right)$, the divergence of X is given by the following:

$$div\left(X\right)={\displaystyle \sum _{i=1}^n}{\epsilon }_{i}g(D_{E_i}X,E_i).$$

(5)

The gradient of a smooth function f on N is the vector field $Df$ characterized by

$$g(Df,X)=X\left(f\right),$$

(6)

for all $X\in \mathfrak{X}\left(N\right)$, and the Hessian of f on N is the symmetric tensor field defined by

$$H^f(X,Y)=g(D_{X}Df,Y),$$

(7)

for all $X,Y\in \mathfrak{X}\left(N\right)$.

The Laplacian of a function f on N is defined as

$$\Delta f=\mathrm{div}\left(Df\right).$$

For $(N,g)$, if there exists a smooth vector field X, and a constant $\kappa$, satisfying the following:

$$Ric+\frac{1}{2}{L}_{X}g=\kappa g,$$

(8)

where $L_{X}g$ represents the Lie derivative of g along X. The structure $\left(N,g,X,\kappa \right)$ is called Ricci soliton.

When the vector field X in a Ricci soliton $\left(N,g,X,\kappa \right)$ is a Killing vector field, the soliton is deemed trivial, and the manifold $(N,g)$ becomes an Einstein. The behavior of $\left(N,g,X,\kappa \right)$ is compellingly categorized based on the constant $\kappa$: it is considered steady when $\kappa =0$, demonstrates a shrinking for $\kappa >0$, and reveals an expanding characteristic when $\kappa <0$. If the vector field X can be represented as the gradient $Df$ of a smooth function f defined on the manifold N, the Ricci soliton $\left(N,g,f,\kappa \right)$ is specifically identified as a gradient Ricci soliton. In this particular scenario, Equation (8) undergoes a transformation into

$$Ric+H^f=\kappa g.$$

(9)

By evaluating the trace of Equation (8), this leads to

$$\mathrm{Scal}+div\left(X\right)=n\kappa .$$

(10)

In case of a gradient Ricci soliton $\left(N,g,f,\kappa \right)$, (10) simplifies to,

$$\mathrm{Scal}+\Delta f=n\kappa .$$

(11)

An affine vector field X on a pseudo-Riemannian manifold $(N,g)$ is defined such that its local flow induces affine transformations. This is represented by the following equation:

$$\left(L_{X}D\right)(Y,Z)=0,$$

(12)

for all $Y,Z\in \mathfrak{X}\left(N\right)$, where $L_X$ denotes the Lie derivative with respect to X. The interaction between $L_X$ and the Levi–Civita connection D is given by the following:

$$\left(L_{X}D\right)(X,Y)=[\zeta ,D_{Y}Z]-D_{[X,Y]}Z-D_Y[X,Z],$$

for any vector fields $Y,Z\in \mathfrak{X}\left(N\right)$. Furthermore, any affine vector field X on a pseudo-Riemannian manifold $(N,g)$ has a constant divergence, indicated by $div\left(X\right)=C$, where C is a constant (refer to [22]).

Now, let $(\overline{N},\overline{g})$ be an (n + 1)-dimensional Riemannian or Lorentzian manifold, $n\ge 3$. Let $(N,g)$ be an oriented Riemannian hypersurface in $(\overline{N},\overline{g})$. The Levi–Civita connections for $(\overline{N},\overline{g})$ and $(N,g)$ are represented by $\overline{D}$ and D, respectively. If $\overline{g}$ is Lorentzian, $(N,g)$ represents a space-like hypersurface.

Choose $U\in \mathfrak{X}{\left(N\right)}^{\top }$ as a unit vector field, which is time-like when $(\overline{N},\overline{g})$ is Lorentzian. The Gauss and Weingarten equations for $(N,g)$ are as follows:

$$\begin{array}{c}\hfill {\overline{D}}_{X}Y=D_{X}Y+\epsilon g\left(\mathbf{S}\left(X\right),Y\right)U,\end{array}$$

(13)

$$\begin{array}{c}\hfill \mathbf{S}\left(X\right)=-{\overline{D}}_{X}U,\end{array}$$

(14)

valid for all $X,Y\in \mathfrak{X}\left(N\right)$, where $\epsilon =\overline{g}(U,U)=\pm 1$, and $\mathbf{S}$ is the shape operator of N concerning U.

The Gauss equation serves as a powerful tool for deriving the curvature tensor R of the hypersurface $(N,g)$ from the curvature tensor $\overline{R}$ associated with the ambient manifold $(\overline{N},\overline{g})$ and the shape operator.

$$R(X,Y)Z={\left(\overline{R}(X,Y)Z\right)}^{\top }+\epsilon \left(g\left(\mathbf{S}\right(Y),Z)\mathbf{S}\left(X\right)-g\left(\mathbf{S}\right(X),Z)\mathbf{S}\left(Y\right)\right),$$

(15)

for any $X,Y,Z\in \mathfrak{X}\left(N\right)$, where ⊤ denotes the tangential part on N.

For the hypersurface $(N,g)$, the mean curvature, denoted as H, plays a crucial role in understanding its geometric properties. It is precisely defined by the equation

$$H=\frac{\epsilon }{n}trace\left(\mathbf{S}\right).$$

(16)

The relationship between the Ricci curvatures $Ric$ of the hypersurface $(N,g)$ and $\overline{Ric}$ of the ambient manifold $(\overline{N},\overline{g})$ is articulated in Equation (17):

$$Ric(X,Y)=\overline{Ric}(X,Y)-\epsilon \overline{g}(\overline{R}(U,X)Y,U)+g(\mathbf{S}\left(X\right),nHY-\epsilon \mathbf{S}\left(Y\right)),$$

(17)

for any vector fields $X,Y\in \mathfrak{X}\left(N\right)$.

Furthermore, by calculating the trace of Equation (17), the significant connection between the scalar curvatures Scal of $(N,g)$ and $\overline{\mathrm{Scal}}$ of $(\overline{N},\overline{g})$ is evident:

$$\mathrm{Scal}=\overline{\mathrm{Scal}}-2\epsilon \overline{Ric}(U,U)+\epsilon \left(n^2{H}^2-{\left|\mathbf{S}\right|}^2\right),$$

(18)

where ${\left|\mathbf{S}\right|}^2=trace\left({\mathbf{S}}^2\right)$.

3. Torse-Forming Vector Fields on Riemannian Manifolds

In this section, we will explore compelling results concerning pseudo-Riemannian manifolds equipped with a torse-forming vector field, alongside a variety of valuable formulas. These formulas empower us to derive significant insights. For an n-dimensional manifold $(N,g)$ having a torse-forming vector field $\zeta$ with its generating form $\mu$ and conformal scalar $\varphi$, we can derive significant conclusions from (5).

$$div\left(\zeta \right)=n\varphi +\mu \left(\zeta \right).$$

(19)

By examining both parts of (19), we find that compact Riemannian manifolds do not have concurrent or recurrent vector fields. Additionally, we can confirm that there are no torqued or closed conformal vector fields linked to a non-zero constant conformal scalar function. This highlights the distinct geometric characteristics of these manifolds.

Define a function f on N by setting $f=\frac{1}{2}g(\zeta ,\zeta )$. If $\eta$ is the dual vector field corresponding to the generating form $\mu$, then it follows

$$\begin{array}{cc}\hfill Y\left(f\right)& =\frac{1}{2}Y·g(\zeta ,\zeta )\hfill \\ & =g(D_Y\zeta ,\zeta )\hfill \\ & =g(\varphi Y+\mu (Y)\zeta ,\zeta )\hfill \\ & =\varphi g(Y,\zeta )+g(Y,\eta )g(\zeta ,\zeta )\hfill \\ & =g(\varphi \zeta +2f\eta ,Y)\hfill \end{array}$$

for all $Y\in \mathfrak{X}\left(N\right)$. It implies that

$$Df=\varphi \zeta +2f\eta .$$

(20)

Lemma 1.

Let ζ be a torse-forming vector field on an n-dimensional pseudo-Riemannian manifold $(N,g)$. Then,

$$\begin{array}{c}\hfill Ric(\zeta ,\zeta )=(n-1)\left(\varphi \mu \right(\zeta )-\zeta (\varphi \left)\right),\end{array}$$

(21)

where μ is the generating form, and ϕ is the conformal scalar.

Proof. 

Assume $\{E_1,\cdots ,E_n\}$ is a parallel local orthonormal frame on the manifold $(N,g)$, and $\eta$ is the dual vector field associated with the generating form $\mu$. It is essential to note that $\mu \left(X\right)$ is defined by the equation $g(X,\eta )$ for every vector field $X\in \mathfrak{X}\left(N\right)$. Considering (1) and (3), we derive the following:

$$\begin{array}{cc}\hfill Ric(\zeta ,\zeta )& ={\displaystyle \sum _{i=1}^n}{\epsilon }_{i}g(R(\zeta ,E_i)\zeta ,E_i)\hfill \\ \hfill & ={\displaystyle \sum _{i=1}^n}{\epsilon }_{i}g(D_{[\zeta ,E_i]}\zeta -D_{\zeta }{D}_{E_i}\zeta +D_{E_i}{D}_{\zeta }\zeta ,E_i)\hfill \\ \hfill & ={\displaystyle \sum _{i=1}^n}{\epsilon }_{i}g(-D_{D_{E_i}\zeta }\zeta -D_{\zeta }(\varphi E_i+\mu \left(E_i\right)\zeta )+D_{E_i}(\varphi \zeta +\mu \left(\zeta \right)\zeta ),E_i)\hfill \\ \hfill & ={\displaystyle \sum _{i=1}^n}{\epsilon }_{i}g(-\varphi D_{E_i}\zeta -\mu \left(E_i\right)D_{\zeta }\zeta -\zeta \left(\varphi \right)E_i-\zeta \left(\mu \left(E_i\right)\right)\zeta -\mu \left(E_i\right)D_{\zeta }\zeta +E_i\left(\varphi \right)\zeta \hfill \\ \hfill & +\varphi D_{E_i}\zeta +E_i\left(\mu \left(\zeta \right)\right)\zeta +\mu \left(\zeta \right)D_{E_i}\zeta ,E_i)\hfill \\ \hfill & ={\displaystyle \sum _{i=1}^n}{\epsilon }_{i}g(-2\mu \left(E_i\right)\varphi \zeta -\mu \left(\zeta \right)\mu \left(E_i\right)\zeta -\zeta \left(\varphi \right)E_i-\zeta \left(\mu \left(E_i\right)\right)\zeta +E_i\left(\varphi \right)\zeta +E_i\left(\mu \left(\zeta \right)\right)\zeta \hfill \\ \hfill & +\varphi \mu \left(\zeta \right)E_i,E_i)\hfill \\ \hfill & ={\displaystyle \sum _{i=1}^n}{\epsilon }_{i}g(-2\mu \left(E_i\right)\varphi \zeta -\zeta \left(\varphi \right)E_i-g(D_{\zeta }\eta ,E_i)\zeta +E_i\left(\varphi \right)\zeta +\varphi \mu \left(E_i\right)\zeta +g(D_{E_i}\eta ,\zeta )\zeta \hfill \\ \hfill & +\varphi \mu \left(\zeta \right)E_i,E_i)\hfill \\ \hfill & =(n-1)\left(\varphi \mu \right(\zeta )-g(D\varphi ,\zeta \left)\right).\hfill \end{array}$$

□

Lemma 2.

Let ζ be a torse-forming vector field on an n-dimensional pseudo-Riemannian manifold $(N,g)$. Then, the Laplacian and the Hessian of the function $f=\frac{1}{2}g(\zeta ,\zeta )$ are given by:

$$\Delta f=n{\varphi }^2+3\varphi \mu \left(\zeta \right)+\zeta \left(\varphi \right)+4f\mu \left(\eta \right)+2fdiv\left(\eta \right),$$

(22)

$$H^f(\zeta ,\zeta )=2f\left(\zeta \left(\varphi \right)+\zeta \left(\mu \left(\zeta \right)\right)+{(\varphi +\mu \left(\zeta \right))}^2\right),$$

(23)

where μ is the generating form, ϕ is the conformal scalar associated with ζ, and η is the dual vector field corresponding to μ.

Proof. 

As $\Delta f=div\left(Df\right)$, we have

$$\begin{array}{cc}\hfill \Delta f& =div(\varphi \zeta +2f\eta )\hfill \\ & =\zeta \left(\varphi \right)+\varphi div\left(\zeta \right)+\eta \left(2f\right)+2fdiv\left(\eta \right)\hfill \\ & =\zeta \left(\varphi \right)+\varphi (n\varphi +\mu (\zeta \left)\right)+2\left(\varphi \mu \right(\zeta )+2fg(\eta ,\eta \left)\right)+2fdiv\left(\eta \right)\hfill \\ & =n{\varphi }^2+3\varphi \mu \left(\zeta \right)+g(D\varphi ,\zeta )+4fg(\eta ,\eta )+2fdiv\left(\eta \right).\hfill \end{array}$$

Also, we have

$$X·g(Df,Y)=X\left(Y\left(f\right)\right)=X·g(D_Y\zeta ,\zeta ),$$

for all $X,Y\in \mathfrak{X}\left(N\right)$, which implies

$$g(D_{X}Df,Y)+g(Df,D_{X}Y)=g(D_X{D}_Y\zeta ,\zeta )+g(D_Y\zeta ,D_X\zeta ).$$

By substituting $X=Y=\zeta$ into the equation above, we can derive the desired expression:

$$H^f(\zeta ,\zeta )=2f\left(\zeta \left(\varphi \right)+\zeta \left(\mu \left(\zeta \right)\right)+{(\varphi +\mu \left(\zeta \right))}^2\right).$$

(24)

□

A torqued vector field within a Riemannian manifold is characterized by the subsequent theorems.

Theorem 1.

Let ζ be a torqued vector field on an n-dimensional Riemannian manifold $(N,g)$, ($n\ge 2$). If $\zeta \left(\varphi \right)\ge 0$, then, $Ric(\zeta ,\zeta )\le 0$ and $H^f(\zeta ,\zeta )\ge 0$.

Proof. 

As $\mu \left(\zeta \right)=0$, we can use (21) and (24) to conclude that:

$$Ric(\zeta ,\zeta )=-(n-1)\zeta \left(\varphi \right)\le 0,$$

and

$$H^f(\zeta ,\zeta )=2f\left(\zeta \left(\varphi \right)+{\varphi }^2\right)\ge 0.$$

□

Theorem 2.

Let ζ be a non-zero torqued vector field on an n-dimensional compact Riemannian manifold $(N,g)$, ($n\ge 2$). If $\zeta \left(\varphi \right)\ge 0$ and η is affine. Then, ζ is parallel.

Proof. 

Integrating (22), it follows that

$${\int }_N\left(n{\varphi }^2+\zeta \left(\varphi \right)+4fg(\eta ,\eta )\right)dV=-{\int }_{N}2fdiv\left(\eta \right)dV.$$

(25)

As $\eta$ is affine, we have $div\left(\eta \right)=C$, where C is a constant. As N is compact, we deduce that $div\left(\eta \right)=0$. Consequently, the expression on the right side of (25) equals zero. As we are assuming $\zeta \left(\varphi \right)\ge 0$, we deduce that $n{\varphi }^2+\zeta \left(\varphi \right)+4fg(\eta ,\eta )=0$, leading to the conclusion that $\varphi =0$ and $\mu =0$. Hence, $\zeta$ is parallel. □

Theorem 3.

Any non-zero torqued vector field with a constant length ζ on a Riemannian manifold $(N,g)$ is parallel.

Proof. 

When the torse-forming vector field $\zeta$ has a constant length. We can deduce from (20) that

$$\varphi \zeta +2f\eta =0,$$

(26)

which yields:

$$\varphi \mu \left(\zeta \right)+2f\mu \left(\eta \right)=0.$$

Given that $\zeta$ is a torqued vector field, which means $\mu \left(\zeta \right)=0$, it follows that $\mu \left(\eta \right)=0$, implying that $\eta =0$, and hence $\mu =0$.

By utilizing this into (26), we deduce that $\varphi =0$, and $\zeta$ is parallel. □

Theorem 4.

A non-zero torqued vector field on a connected Riemannian manifold is necessarily closed conformal.

Proof. 

Assume that $\zeta$ is a non-zero torqued vector field on a connected Riemannian manifold $(N,g)$. Denote $\eta$ as the dual vector field corresponding to the generating form $\mu$, where $\mu \left(X\right)=g(X,\eta )$ for all $X\in \mathfrak{X}\left(N\right)$. As $\mu \left(\zeta \right)=0$, then we obtain

$$\begin{array}{cc}\hfill 0=L_{\zeta }\mu \left(\zeta \right)& =L_{\zeta }g(\zeta ,\eta )\hfill \\ & =g(D_{\zeta }\zeta ,\eta )+g(D_{\eta }\zeta ,\zeta )\hfill \\ & =\varphi \mu \left(\zeta \right)+\mu \left(\eta \right)g(\zeta ,\zeta )\hfill \\ & =g(\eta ,\eta )g(\zeta ,\zeta ).\hfill \end{array}$$

Because $\zeta$ is non-zero, we obtain $\eta =0$. Hence, $\mu =0$, meaning $\zeta$ is a closed conformal. □

Theorem 5.

Let ζ be a torqued vector field on an n-dimensional compact Riemannian manifold $(N,g)$, ($n\ge 2$). If $Ric(\zeta ,\zeta )\le 0$ and η is affine, then ζ is parallel.

Proof. 

From (21), we obtain:

$$\zeta \left(\varphi \right)\ge 0.$$

As $\eta$ is affine, we have $div\left(\eta \right)=C$, where C is a constant. As N is compact, we deduce that $div\left(\eta \right)=0$.

It follows by (22) that:

$$\Delta f=n{\varphi }^2+\zeta \left(\varphi \right)+4f\mu \left(\eta \right)\ge 0.$$

(27)

As N is compact, we deduce that $\varphi =0$, and $\mu \left(\eta \right)=0$, which means that $\mu =0$. Hence, $\zeta$ is parallel. □

Inspired by the proof of Theorem 5, we derive the following result.

Corollary 1.

Let ζ be a torqued vector field on an n-dimensional compact Riemannian manifold $(N,g)$, ($n\ge 2$). If $\zeta \left(\varphi \right)\ge 0$ and η is affine, then ζ is parallel. In particular, if ϕ is constant along the integral curves of ζ and η is affine, then ζ must be parallel.

A later result demonstrates that the sign of $Ric(\zeta ,\zeta )$ is notably significant for a closed conformal vector field $\zeta$.

Theorem 6.

Let ζ be a closed conformal vector field on an n-dimensional compact Riemannian manifold $(N,g)$, ($n\ge 2$). If $Ric(\zeta ,\zeta )\le 0$, then ζ is parallel.

Proof. 

As $\zeta$ is a closed conformal vector field (i.e., $\mu =0$), then by combining (21) and (22), we obtain

$$\Delta f=n{\varphi }^2-\frac{1}{n-1}Ric(\zeta ,\zeta ).$$

(28)

Integrating both sides of the given equation results in the following:

$${\int }_{N}Ric(\zeta ,\zeta )dV=n(n-1){\int }_N{\varphi }^{2}dV.$$

(29)

As $Ric(\zeta ,\zeta )\le 0$, we conclude that $\varphi =0$, implying that $\zeta$ remains parallel. □

The result arises from the existence of a recurrent or concurrent vector field on a Riemannian manifold.

Theorem 7.

Let ζ be a recurrent or concurrent vector field on an n-dimensional Riemannian manifold $(N,g)$, $n\ge 3$. Then, $Ric(\zeta ,\zeta )=0$ and $H^f(\zeta ,\zeta )\ge 0$.

Proof. 

If $\zeta$ is concurrent, which means $\varphi$ is constant and $\mu =0$, or recurrent (i.e., $\varphi =0$ and $\mu$ is constant), then (21) becomes $Ric(\zeta ,\zeta )=0$.

We also deduce from (24) that is both cases where $\zeta$ is concurrent or recurrent that $H^f(\zeta ,\zeta )=0$ or $H^f(\zeta ,\zeta )\ge 0$, respectively. □

4. Ricci Solitons on Riemannian Hypersurfaces Generated by Torse-Forming Vector Fields in Riemannian and Lorentzian Manifolds

Let $(\overline{N},\overline{g})$ be an (n + 1)-dimensional Riemannian or Lorentzian manifold, $n\ge 3$. Let $(N,g)$ be an oriented Riemannian hypersurface in $(\overline{N},\overline{g})$. The Levi–Civita connections for $(\overline{N},\overline{g})$ and $(N,g)$ are represented by $\overline{D}$ and D, respectively.

For a torse-forming vector field $\overline{\zeta }$ on $\overline{N}$, that is $\overline{\zeta }$ satisfies

$${\overline{D}}_X\overline{\zeta }=\varphi X+\mu \left(X\right)\overline{\zeta },$$

(30)

for all $X\in \mathfrak{X}\left(\overline{N}\right)$, $\varphi$ is a smooth function on $\overline{N}$ (conformal scalar) and $\mu$ denotes a one-form (generating form). In the case of a Lorentzian manifold $(\overline{N},\overline{g})$, $\overline{\zeta }$ is time-like.

Choose $U\in \mathfrak{X}{\left(N\right)}^{\top }$ as a unit vector field, which is considered time-like if $(\overline{N},\overline{g})$ is Lorentzian.

Let $\eta$ be a dual vector field corresponding to $\mu$ with respect to $\overline{g}$, that is $\mu \left(X\right)=\overline{g}(X,\eta )$, for all $X\in \mathfrak{X}\left(\overline{N}\right)$. Let $\zeta$ denote the restriction of $\overline{\zeta }$ to N, as represented by:

$$\zeta ={\zeta }^{\top }+\epsilon \theta U,$$

(31)

where $\theta =\overline{g}(\zeta ,U)$, ${\zeta }^{\top }$ is the tangential part of $\zeta$, and $\epsilon =\overline{g}(U,U)=\pm 1$. In the context of a Lorentzian manifold $(\overline{N},\overline{g})$, the reverse Cauchy–Schwarz inequality implies that $\theta <0$ everywhere.

Given that $\zeta$ is a torse-forming vector field, it follows by applying (13), (14), (30) and (31) that

$$D_X{\zeta }^{\top }=\varphi X+\epsilon \theta \mathbf{S}\left(X\right)+\mu \left(X\right){\zeta }^{\top },$$

(32)

holds for any $X\in \mathfrak{X}\left(N\right)$. Additionally, we have

$$D\theta =-\mathbf{S}\left({\zeta }^{\top }\right)+\theta {\eta }^{\top },$$

(33)

where ${\eta }^{\top }$ denotes the tangential part of $\eta$. Recall that D is the Levi–Civita connection on $(N,g)$, and $\mathbf{S}$ represents the shape operator. From (32), it is straightforward to derive

$$div\left({\zeta }^{\top }\right)=n(\varphi +\theta H)+\mu \left({\zeta }^{\top }\right).$$

(34)

Define T and $\overline{T}$ as the Ricci operators for $(N,g)$ and $(\overline{N},\overline{g})$, respectively. These operators relate to the Ricci curvatures $Ric$ and $\overline{Ric}$ of $(N,g)$ and $(\overline{N},\overline{g})$ through the equations:

$$Ric(X,Y)=g\left(T\right(X),Y),$$

and

$$\overline{Ric}(X,Y)=\overline{g}(\overline{T}\left(X\right),Y).$$

As $\overline{g}(\overline{R}(U,X)U,U)=0$ for all $X\in \mathfrak{X}\left(N\right)$, we can define the normal Jacobi operator${\mathfrak{R}}_U:TN\to TN$ as the operator defined by:

$${\mathfrak{R}}_U\left(X\right)=\overline{R}(U,X)U,$$

(35)

for all $X\in \mathfrak{X}\left(N\right)$.

Also, we define $\alpha :\mathfrak{X}\left(N\right)⟶\mathrm{Span}\{{\zeta }^{\top },{\eta }^{\top }\}$ by

$$\alpha \left(X\right)=\mu \left(X\right){\zeta }^{\top }+\beta \left(X\right){\eta }^{\top },$$

(36)

for every $X\in \mathfrak{X}\left(N\right)$, where $\beta \left(X\right)=\overline{g}(\zeta ,X)$ for every $X\in \mathfrak{X}\left(\overline{N}\right)$.

The following theorem examines the essential conditions for the tangential component ${\zeta }^{\top }$ to generate a Ricci soliton on the Riemannian hypersurface $(N,g)$.

Theorem 8.

Let $(N,g)$ be an oriented Riemannian hypersurface of an (n + 1)-dimensional Riemannian or Lorentzian manifold $(\overline{N},\overline{g})$, with $n\ge 3$, which admits a torse-forming vector field $\overline{\zeta }$ (time-like if $(\overline{N},\overline{g})$ is Lorentzian) with conformal scalar ϕ and generating form μ. With the notations above, $(N,g,{\zeta }^{\top },\kappa )$ is a Ricci soliton if and only if the following equation holds:

$$\overline{T}-\epsilon {\mathfrak{R}}_U=\epsilon {\mathbf{S}}^2-(\epsilon \theta +nH)\mathbf{S}+(\kappa -\varphi )I-\frac{1}{2}\alpha .$$

(37)

Proof. 

By utilizing (32), we derive the following expression:

$$\begin{array}{cc}\hfill L_{{\zeta }^{\top }}g(X,Y)& =g(D_X{\zeta }^{\top },Y)+g(X,D_Y{\zeta }^{\top })\hfill \\ & =g(\varphi X+\epsilon \theta \mathbf{S}\left(X\right)+\mu \left(X\right){\zeta }^{\top },Y)+g(X,\varphi Y+\epsilon \theta \mathbf{S}\left(Y\right)+\mu \left(Y\right){\zeta }^{\top })\hfill \\ & =2\varphi g(X,Y)+2\epsilon \theta g\left(\mathbf{S}\right(X),Y)+\mu \left(X\right)\beta \left(Y\right)+\mu \left(Y\right)\beta \left(X\right),\hfill \end{array}$$

for all $X,Y\in \mathfrak{X}\left(N\right)$.

Given $(N,g,{\zeta }^{\top },\kappa )$ as a Ricci soliton, the equation holds

$$Ric+\frac{1}{2}{L}_{{\zeta }^{\top }}g=\kappa g,$$

(38)

where $\kappa$ is a constant.

Therefore, for all $X,Y\in \mathfrak{X}\left(N\right)$, we have:

$$\begin{array}{cc}\hfill \kappa g(X,Y)& =g(T\left(X\right),Y)+\varphi g(X,Y)+\epsilon \theta g(\mathbf{S}\left(X\right),Y)+\frac{1}{2}(\mu \left(X\right)\beta \left(Y\right)+\mu \left(Y\right)\beta \left(X\right))\hfill \\ & =g(T\left(X\right),Y)+\varphi g(X,Y)+\epsilon \theta g(\mathbf{S}\left(X\right),Y)+\frac{1}{2}\mu \left(X\right)g({\zeta }^{\top },Y)\hfill \\ & +\frac{1}{2}g({\eta }^{\top },Y)\beta \left(X\right)\hfill \\ & =g(T\left(X\right)+\varphi X+\epsilon \theta \mathbf{S}\left(X\right)+\frac{1}{2}\mu \left(X\right){\zeta }^{\top }+\frac{1}{2}\beta \left(X\right){\eta }^{\top },Y)\hfill \end{array}$$

Hence, T can be expressed as follows:

$$T\left(X\right)=(\kappa -\varphi )X-\epsilon \theta \mathbf{S}\left(X\right)-\frac{1}{2}(\mu \left(X\right){\zeta }^{\top }+\beta \left(X\right){\eta }^{\top }),$$

(39)

for all $X\in \mathfrak{X}\left(N\right)$.

Additionally, (17) provides the following representation:

$$T\left(X\right)=\overline{T}\left(X\right)-\epsilon \overline{R}(U,X)U+nH\mathbf{S}\left(X\right)-\epsilon {\mathbf{S}}^2\left(X\right).$$

(40)

By setting (39) and (40) equal to each other, we can derive (37). The converse is obvious.    ☐

As a consequence of Theorem 8, we have the following:

Theorem 9.

For an oriented Riemannian hypersurface $(N,g)$ of an (n + 1)-dimensional Riemannian or Lorentzian manifold $(\overline{N},\overline{g})$, with $n\ge 3$, which admits a torse-forming vector field $\overline{\zeta }$ (time-like if $(\overline{N},\overline{g})$ is Lorentzian) with conformal scalar ϕ and generating form μ. Given the assumptions and notions outlined in Theorem 8. For $(N,g,{\zeta }^{\top },\kappa )$ as a Ricci soliton, the following equation holds:

$${\left|\mathbf{S}\right|}^2-n{H}^2+\epsilon n(\kappa -\varphi -\theta H)-\epsilon \mu \left({\zeta }^{\top }\right)-\epsilon n(n-1)(\epsilon H^2+\overline{C})=0,$$

(41)

where ${\left|\mathbf{S}\right|}^2=trace\left({\mathbf{S}}^2\right)$.

Proof. 

Given that $(\overline{N},\overline{g})$ is of constant sectional curvature $\overline{C}$, then from Corollary 3.43 in [21], we have

$${\mathfrak{R}}_U\left(X\right)=\overline{C}\left(\overline{g}(U,U)X-\overline{g}(U,X)U\right)=\epsilon \overline{C}X,$$

(42)

for all $X\in \mathfrak{X}\left(N\right)$.

Also, by Corollary 3.43 in [21] and (3), we have $\overline{T}=n\overline{C}I$.

It follows that (37) becomes

$$\epsilon {\mathbf{S}}^2-(\epsilon \theta +nH)\mathbf{S}+(\kappa -\varphi -(n-1)\overline{C})I-\frac{1}{2}\alpha =0.$$

Evaluating the trace of the preceding equation yields

$${\epsilon \left|\mathbf{S}\right|}^2-\epsilon n(\epsilon \theta +nH)H+n\left(\kappa -\varphi -(n-1)\overline{C}\right)-\mu \left({\zeta }^{\top }\right)=0,$$

or equivalently,

$${\left|\mathbf{S}\right|}^2-n{H}^2+\epsilon n(\kappa -\varphi -\theta H)-\epsilon \mu \left({\zeta }^{\top }\right)-n(n-1)(H^2+\epsilon \overline{C})=0.$$

(43)

□

According to (10) and (34), we can derive that

$$\mathrm{Scal}=n(\kappa -\varphi -\theta H)-\mu \left({\zeta }^{\top }\right)$$

(44)

Consequently, Theorem 9 can be reformulated to incorporate the scalar curvature of hypersurfaces.

Corollary 2.

Let $(N,g)$ be an oriented Riemannian hypersurface of an (n + 1)-dimensional Riemannian or Lorentzian manifold $(\overline{N},\overline{g})$, with $n\ge 3$, which admits a torse-forming vector field $\overline{\zeta }$ (time-like if $(\overline{N},\overline{g})$ is Lorentzian). Given the assumptions and notions outlined in Theorem 8, if $(N,g,{\zeta }^{\top },\kappa )$ is a Ricci soliton, this equation is therefore fulfilled:

$${\left|\mathbf{S}\right|}^2-n{H}^2+\epsilon \mathrm{Scal}-\epsilon n(n-1)(\epsilon H^2+\overline{C})=0.$$

(45)

When discussing space forms, it is essential to consider the following insightful characterization of compact hypersurfaces.

Theorem 10.

Let $(N,g)$ be an oriented compact Riemannian hypersurface of an (n + 1)-dimensional Riemannian or Lorentzian manifold $(\overline{N},\overline{g})$ of a constant sectional curvature $\overline{C}$, with $n\ge 3$, which admits a torse-forming vector field $\overline{\zeta }$ that we assume to be time-like if $\overline{N}$ is Lorentzian. Then, if $(N,g,{\zeta }^{\top },\kappa )$ is non-trivial, and $\epsilon \kappa \ge (n-1)(H^2+\epsilon \overline{C})$, then $(N,g)$ is a sphere, with sectional curvature $C=\epsilon H^2+\overline{C}$. In particular, if $(\overline{N},\overline{g})$ is Riemannian, the Ricci soliton shrinks, whereas if $(\overline{N},\overline{g})$ is Lorentzian, the Ricci soliton is trivial.

Proof. 

By integration (41) and taking into account (34), we obtain

$${\int }_N\left(n{H}^2-{\left|\mathbf{S}\right|}^2\right)dV=n{\int }_N\left(\epsilon \kappa -(n-1)(H^2+\epsilon \overline{C})\right)dV.$$

(46)

Given that $n{H}^2-{\left|\mathbf{S}\right|}^2\le 0$, by the Cauchy–Schwarz inequality, and the assumption $\epsilon \kappa \ge (n-1)(H^2+\epsilon \overline{C})$, we deduce that $(N,g)$ is totally umbilical.

According to Lemma 4.35 from [21], the mean curvature H is constant, and the sectional curvature of $(N,g)$$C=\overline{C}+\epsilon H^2$ is constant. As $(N,g)$ is compact, it is isometric to a sphere, leading to $\epsilon \kappa >0$. In the Riemannian case of $(\overline{N},\overline{g})$, this implies $\kappa >0$, signifying a shrinking Ricci soliton. Conversely, if $(\overline{N},\overline{g})$ is Lorentzian, then $\kappa <0$, which means that the Ricci soliton is expanding. In this situation, as N is compact, the Ricci soliton must be trivial (see [23]). □

When the hypersurface is minimal or maximal, we obtain the subsequent results.

Corollary 3.

Let $(N,g)$ be an oriented compact minimal hypersurface of an (n + 1)-dimensional Riemannian manifold $(\overline{N},\overline{g})$ with a constant sectional curvature $\overline{C}\le 0$, with $n\ge 3$, and which admits a torse-forming vector field $\overline{\zeta }$. If $(N,g,{\zeta }^{\top },\kappa )$ is a Ricci soliton, then it must be trivial.

Proof. 

As $(N,g)$ is minimal, the mean curvature $H=0$. From the proof of Theorem 10, we have

$${\int }_N{\left|\mathbf{S}\right|}^{2}dV=n{\int }_N\left((n-1)\overline{C}-\kappa \right)dV=n\left((n-1)\overline{C}-\kappa \right)Vol\left(N\right),$$

(47)

with $Vol\left(N\right)$ as the volume of N. As ${\left|\mathbf{S}\right|}^2\ge 0$, it follows that $\kappa \le (n-1)\overline{C}$. If we assume $\overline{C}\le 0$, we deduce that $\kappa \le 0$. Therefore, the Ricci soliton is either steady or expanding. As N is compact, the Ricci soliton must be trivial (see [23]). □

Corollary 4.

Let $(N,g)$ be an oriented compact maximal hypersurface of an (n + 1)-dimensional Lorentzian manifold $(\overline{N},\overline{g})$ with a constant sectional curvature $\overline{C}>0$, with $n\ge 3$, and which admits a torse-forming vector field $\overline{\zeta }$. If $(N,g,{\zeta }^{\top },\kappa )$ is a Ricci soliton, it must be shrinking.

Proof. 

As $(N,g)$ is maximal, the mean curvature $H=0$. From the proof of Theorem 10, we have

$${\int }_N{\left|\mathbf{S}\right|}^{2}dV=n{\int }_N\left(\kappa -(n-1)\overline{C}\right)dV=n\left(\kappa -(n-1)\overline{C}\right)Vol\left(N\right),$$

(48)

with $Vol\left(N\right)$ as the volume of N. As ${\left|\mathbf{S}\right|}^2\ge 0$, we deduce that $\kappa \ge (n-1)\overline{C}$. If we assume $\overline{C}>0$, it follows that $\kappa >0$. Therefore, the Ricci soliton is shrinking. □

Corollary 5.

Let $(N,g)$ be an oriented compact Riemannian hypersurface of an (n + 1)-dimensional Riemannian or Lorentzian manifold $(\overline{N},\overline{g})$ of constant sectional curvature $\overline{C}$, with $n\ge 3$, which admits a torse-forming vector field $\overline{\zeta }$ that we assume to be time-like if $\overline{N}$ is Lorentzian. Assume that $H^2+\epsilon \overline{C}\le 0$. A non-trivial Ricci soliton $(N,g,{\zeta }^{\top },\kappa )$ becomes trivial if $\epsilon \kappa \ge (n-1)(H^2+\epsilon \overline{C})$ and $(\overline{N},\overline{g})$ is Riemannian, whereas it is shrinking if $(\overline{N},\overline{g})$ is Lorentzian.

Proof. 

According to Theorem 10, we can assert that $\epsilon \kappa =(n-1)(H^2+\epsilon \overline{C})$. Thus, under the assumption that $H^2+\epsilon \overline{C}\le 0$, it leads to the conclusion that $\epsilon \kappa \le 0$. The situation where the compactness of N excludes $\kappa =0$. As a result, with $\epsilon \kappa <0$. Thus, we have two cases: in the case of Riemannian (i.e., for $\epsilon =1$), the Ricci soliton expands, as N is compact; from the result in [23], the Ricci soliton must be trivial. In the case of Lorentzian (i.e., for $\epsilon =-1$), the soliton shrinks. □

Theorem 11.

Let $(N,g)$ be an oriented connected and geodesically complete Riemannian hypersurface of an (n + 1)-dimensional Riemannian or Lorentzian manifold $(\overline{N},\overline{g})$ of constant sectional curvature $\overline{C}$, with $n\ge 3$, which admits a torse-forming vector field $\overline{\zeta }$ that we assume to be time-like if $\overline{N}$ is Lorentzian. Assume that $\epsilon \mathit{Scal}\ge n(n-1)(H^2+\epsilon \overline{C})$. If $(N,g,{\zeta }^{\top },\kappa )$ is a Ricci soliton, then the mean curvature and sectional curvature of $(N,g)$ are constant. $(N,g)$ is isometric to one of three spaces: the Euclidean space ${\mathbf{R}}^n$, the sphere ${\mathbf{S}}^n\left(C\right)$, or the hyperbolic space ${\mathbf{H}}^n\left(C\right)$.

Proof. 

Given that the Cauchy–Schwarz inequality ${\left|\mathbf{S}\right|}^2-n{H}^2\ge 0$ holds, and as $\epsilon \mathrm{Scal}\ge n(n-1)(H^2+\epsilon \overline{C})$, we deduce from (45) that $(N,g)$ is totally umbilical.

According to Lemma 4.35 in [21], the mean curvature H and the sectional curvature of $(N,g)$ are both constant, the latter given by $C=\overline{C}+\epsilon H^2$.

According to Hopf’s result, if $C=0$, the manifold $(N,g)$ is Euclidean ${\mathbf{R}}^n$; for positive C, $(N,g)$ is a sphere ${\mathbf{S}}^n\left(C\right)$; and for negative C, it is hyperbolic ${\mathbf{H}}^n\left(C\right)$. □

5. Conclusions

Torse-forming vector fields are essential in several areas of geometry and physics, notably in relativity, cosmology, and submanifold theory. Their rich structure and broad applicability make them a subject of significant interest.

This paper offers a detailed investigation into torse-forming vector fields on pseudo-Riemannian manifolds, highlighting their fundamental role in studying Ricci solitons on Riemannian and spacelike hypersurfaces within both Riemannian and Lorentzian geometries, respectively.

The interplay between torse-forming vector fields and the development of Ricci solitons is a key focus, opening up new directions for future research.

Beyond extending classical Ricci soliton theory, this work provides novel insights into Riemannian and spacelike hypersurfaces subject to some specific geometric constraints in spaces of constant sectional curvature.

The results underscore the value of torse-forming vector fields in shaping the geometric character of such hypersurfaces. As this field evolves, ongoing exploration of the relationship between torse-forming vector fields and Ricci solitons is expected to further illuminate the geometric and physical structures of pseudo-Riemannian manifolds.