Ricci Vector Fields Revisited

Abstract

We continue studying the $\sigma$-Ricci vector field $\mathbf{u}$ on a Riemannian manifold $(N^m,g)$, which is not necessarily closed. A Riemannian manifold with Ricci operator T, a Coddazi-type tensor, is called a T-manifold. In the first result of this paper, we show that a complete and simply connected T-manifold $(N^m,g)$, $m>1$, of positive scalar curvature $\tau$, admits a closed $\sigma$-Ricci vector field $\mathbf{u}$ such that the vector $\mathbf{u}-\nabla \sigma$ is an eigenvector of T with eigenvalue $\tau m^{-1}$, if and only if it is isometric to the m-sphere $S_{\alpha }^m$. In the second result, we show that if a compact and connected T-manifold $(N^m,g)$, $m>2$, admits a $\sigma$-Ricci vector field $\mathbf{u}$ with $\sigma \ne 0$ and is an eigenvector of a rough Laplace operator with the integral of the Ricci curvature $Ric\left(\mathbf{u},\mathbf{u}\right)$ that has a suitable lower bound, then $(N^m,g)$ is isometric to the m-sphere $S_{\alpha }^m$, and the converse also holds. Finally, we show that a compact and connected Riemannian manifold $(N^m,g)$ admits a $\sigma$-Ricci vector field $\mathbf{u}$ with $\sigma$ as a nontrivial solution of the static perfect fluid equation, and the integral of the Ricci curvature $Ric\left(\mathbf{u},\mathbf{u}\right)$ has a lower bound depending on a positive constant $\alpha$, if and only if $(N^m,g)$ is isometric to the m-sphere $S_{\alpha }^m$.

1. Introduction

In a recent paper, (cf. [1]), a $\sigma$-Ricci vector field (abbreviated as $\sigma$-RVF) $\mathbf{u}$ on a m-Riemannian manifold $\left(N^m,g\right)$ is introduced, being defined by

$$\frac{1}{2}{\pounds }_{\mathbf{u}}g=\sigma Ric,$$

(1)

where ${\pounds }_{\mathbf{u}}g$ is the Lie derivative of the metric g with respect to $\mathbf{u}$, $\sigma$ is a smooth function and $Ric$ is the Ricci tensor of $\left(N^m,g\right)$. A $\sigma$-RVF is a generalization of conformal vector fields (known for their utility in studying geometry and relativity), on Einstein manifolds (see [1,2,3,4,5,6,7,8,9,10,11]). Moreover, it represents a Killing vector field, which is known to have a great influence on the geometry as well as topology on which it lives (see [12,13,14,15]). Apart from these generalizations, a $\sigma$-RVF is a particular form of potential field of generalized solitons considered in [16,17,18]. Note that a 1-RVF $\mathbf{u}$ on a m-Riemannian manifold $\left(N^m,g\right)$ is a stable Ricci soliton $\left(N^m,g,\mathbf{u},0\right)$ (see [19]). Indeed, in [1], it has been observed that a $\sigma$-RVF on $\left(N^m,g\right)$ is a stable solution of the generalized Ricci flow (or a $\sigma$-Ricci flow),

$${\partial }_{t}g=2\sigma Ric,g\left(0\right)=g,$$

(2)

of the form $g\left(t\right)=\rho \left(t\right){\phi }_t^{*}\left(g\right)$, where ${\phi }_t:N^m\to N^m$ is a 1-parameter family of diffeomorphisms generated by the vector fields $\mathbf{U}\left(t\right)$ and $\rho \left(t\right)$ is a scale factor, under the initial conditions $\rho \left(0\right)=1$, $\stackrel{.}{\rho }\left(0\right)=0$, $\mathbf{U}\left(0\right)=\mathbf{u}$ and ${\phi }_0=id$.

In [1], a closed $\sigma$-RVF $\mathbf{u}$, with $\sigma \ne 0$, on a compact and connected m-Riemannian manifold $\left(N^m,g\right)$, $m>2$, of nonzero scalar curvature is used with an appropriate lower bound on the integral of the Ricci curvature $Ric\left(\mathbf{u},\mathbf{u}\right)$ to find a characterization of the m-sphere $S^m\left(c\right)$. Moreover, in [1], a closed $\sigma$-RVF $\mathbf{u}$ on a complete and simply connected m-Riemannian manifold $\left(N^m,g\right)$, $m>2$, of positive scalar curvature, is used, where the function $\sigma$ is a nontrivial solution of the Fischer–Marsden equation (cf. [20]) with an appropriate upper bound on the length $∥\nabla \mathbf{u}∥$ of the covariant derivative of $\mathbf{u}$, to find another characterization of the sphere $S^m\left(c\right)$.

The Ricci operator T of a Riemannian manifold $\left(N^m,g\right)$ is a symmetric operator defined by

$$Ric\left(E,F\right)=g\left(T\left(E\right),F\right),E,F\in \Gamma \left(N^m\right),$$

where $\Gamma \left(N^m\right)$ is a space of vector fields on $N^m$. A Riemannian manifold $\left(N^m,g\right)$ is said to be a T-manifold, if the Ricci operator T is a Codazzi tensor, i.e., it satisfies

$$\left(D_{E}T\right)\left(F\right)=\left(D_{F}T\right)\left(E\right),E,F\in \Gamma \left(N^m\right),$$

(3)

where D is the Riemannian connection on $\left(N^m,g\right)$. It is worth noting that a T-manifold $\left(N^m,g\right)$ has a constant scalar curvature.

In this article, we are interested in studying the geometry of $\left(N^m,g\right)$ equipped with a $\sigma$-RVF $\mathbf{u}$. In the first result, we consider a T-manifold $\left(N^m,g\right)$ that possesses a closed $\sigma$-RVF $\mathbf{u}$ and we observe that, in this case, the vector field $\mathbf{u}-\nabla \sigma$ has a special role to play in shaping the geometry of the T-manifold $\left(N^m,g\right)$. It is shown that if the scalar curvature $\tau$ of a compact T-manifold $\left(N^m,g\right)$ is positive (note that $\tau$ is a constant for a T-manifold) and the vector field $\mathbf{u}-\nabla \sigma$ satisfies

$$T\left(\mathbf{u}-\nabla \sigma \right)=\frac{\tau }{m}\left(\mathbf{u}-\nabla \sigma \right),$$

then $\left(N^m,g\right)$ is isometric to the m-sphere $S_c^m$ of constant curvature c, where $\tau =m(m-1)c$, and the converse also holds (cf. Theorem 1).

Then, we concentrate on a $\sigma$-RVF $\mathbf{u}$ on $\left(N^m,g\right)$ that is not necessarily closed. In this case, the 1-form $\beta$ dual to $\mathbf{u}$ gives rise to a skew symmetric operator $\Psi :\Gamma \left(N^m\right)\to \Gamma \left(N^m\right)$ defined by

$$g\left(\Psi \left(E\right),F\right)=\frac{1}{2}d\beta \left(E,F\right),E,F\in \Gamma \left(N^m\right),$$

and we call the operator $\Psi$ the associated operator of the $\sigma$-RVF $\mathbf{u}$. In the second result of this paper, we consider a compact and connected T-manifold $\left(N^m,g\right)$ with scalar curvature $\tau =m(m-1)c$ that possesses a $\sigma$-RVF $\mathbf{u}$, $\sigma \ne 0$, with associated operator $\Psi$ satisfying

$$\Delta \mathbf{u}=-c\mathbf{u},{\int }_{N^m}Ric\left(\mathbf{u},\mathbf{u}\right)\ge {\int }_{N^m}\left[\frac{m-1}{m}{\tau }^2{\sigma }^2+{∥\Psi ∥}^2\right],$$

which necessarily implies that $\left(N^m,g\right)$ is isometric to the m-sphere $S_c^m$ of constant curvature c, and the converse is also true (cf. Theorem 2), where $\Delta$ is the rough Laplace operator acting on vector fields on $\left(N^m,g\right)$.

Recall the differential equation on a Riemannian manifold $\left(N^m,g\right)$ considered by Obata (cf. [18,21]), namely

$$Hess\left(\sigma \right)=-c\sigma g,$$

(4)

where $\sigma$ is a non-constant smooth function, c is a positive constant and $Hess\left(\sigma \right)$ is the Hessian of $\sigma$ defined by

$$Hess\left(\sigma \right)(E,F)=g\left(D_E\nabla \sigma ,F\right),E,F\in \Gamma \left(N^m\right).$$

It is known that a complete, simply connected $\left(N^m,g\right)$ admits a nontrivial solution of (4) if and only if $\left(N^m,g\right)$ is isometric to the sphere $S_c^m$ (cf. [18,21]).

There is yet another important differential equation on a Riemannian manifold $\left(N^m,g\right)$ (cf. [7] and references therein), given by

$$\sigma Ric-Hess\left(\sigma \right)=\frac{1}{m}\left(\tau \sigma -\Delta \sigma \right)g,$$

(5)

known as the static fluid equation, where $\Delta \sigma$ is the Laplacian of $\sigma$ with respect to the metric g. A Riemannian manifold $\left(N^m,g\right)$ that admits a nontrivial solution of the static fluid equation is called a static space. Note that under the additional assumption

$$\Delta \sigma =-\frac{\tau }{m-1}\sigma ,$$

the static fluid equation reduces to the Fischer–Marsden equation (cf. [20])

$$\left(\Delta \sigma \right)g+\sigma Ric=Hess\left(\sigma \right).$$

(6)

In the last result of this paper, we show that a compact and connected Riemannian manifold $\left(N^m,g\right)$ with scalar curvature $\tau$ possessing a $\sigma$-RVF $\mathbf{u}$ with associated operator $\Psi$ and the function $\sigma$ is a nontrivial solution of the static perfect fluid Equation (5); furthermore, for a positive constant c, the following inequality holds:

$${\int }_{N^m}Ric\left(\mathbf{u},\mathbf{u}\right)\ge {\int }_{N^m}\left[\frac{m-1}{m}{\tau }^2{\sigma }^2+\frac{1}{m}{\left(\Delta \sigma +nc\sigma \right)}^2+{∥\Psi ∥}^2\right],$$

which necessarily implies that $\left(N^m,g\right)$ is isometric to the sphere $S_c^m$, and the converse is also true (cf. Theorem 3).

2. Preliminaries

For a $\sigma$-RVF $\mathbf{u}$ on an m-dimensional Riemannian manifold $\left(N^m,g\right)$, we let $\beta$ be the 1-form dual to $\mathbf{u}$, i.e.,

$$\beta \left(E\right)=g\left(\mathbf{u},E\right),E\in \Gamma \left(N^m\right).$$

(7)

Then, we have the associated operator $\Psi$ satisfying

$$d\beta \left(E,F\right)=\frac{1}{2}g\left(\Psi \left(E\right),F\right),E,F\in \Gamma \left(N^m\right),$$

(8)

which shows that $\Psi$ is a skew symmetric operator. Using Equations (1) and (8), we obtain the following expression for the covariant derivative ${\nabla }_E\mathbf{u}$

$$D_E\mathbf{u}=\sigma T\left(E\right)+\Psi \left(E\right),E\in \Gamma \left(N^m\right).$$

(9)

where T is the Ricci operator defined by

$$Ric\left(E,F\right)=g\left(T\left(E\right),F\right),E,F\in \Gamma \left(N^m\right).$$

On employing the following expression for the curvature tensor field R of $\left(N^m,g\right)$,

$$R\left(E,F\right)G=\left[D_E,D_F\right]G-D_{[E,F]}G,E,F,G\in \Gamma \left(N^m\right),$$

with Equation (9), we obtain

$$\begin{array}{cc}\hfill R\left(E,F\right)\mathbf{u}& =E\left(\sigma \right)T\left(F\right)-F\left(\sigma \right)T\left(E\right)+\rho \left(\left(D_{E}T\right)\left(F\right)-\left(D_{F}T\right)\left(E\right)\right)\hfill \\ \hfill & +\left(D_E\Psi \right)\left(F\right)-\left(D_F\Psi \right)\left(E\right),\hfill \end{array}$$

(10)

for any $E,F\in \Gamma \left(N^m\right)$, where

$$\left(D_{E}T\right)\left(F\right)=D_{E}T\left(F\right)-T\left(D_{E}F\right).$$

The scalar curvature $\tau$ of $\left(N^m,g\right)$ is given by

$$\tau =\sum _{\alpha =1}^{m}g\left(T\left(E_{\alpha }\right),E_{\alpha }\right),$$

where $\left\{E_1,\dots ,E_m\right\}$ is a local frame on $N^m$. The Ricci tensor is given by

$$Ric\left(E,F\right)=\sum _{\alpha =1}^{m}g\left(R\left(E_{\alpha },E\right)F,E_{\alpha }\right),$$

and employing it in Equation (10), we conclude

$$\begin{array}{cc}\hfill Ric\left(F,\mathbf{u}\right)& =Ric\left(F,\nabla \sigma \right)-\tau F\left(\sigma \right)+\sigma g\left(F,\sum _{\alpha =1}^m\left({\nabla }_{F_{\alpha }}T\right)\left(F_{\alpha }\right)\right)\hfill \\ \hfill & -\rho Y\left(\tau \right)-g\left(F,\sum _{\alpha =1}^m\left({\nabla }_{F_{\alpha }}\Psi \right)\left(F_{\alpha }\right)\right),\hfill \end{array}$$

(11)

where $\nabla \sigma$ is the gradient of $\sigma$ and we have used the symmetry of the Ricci operator T and the skew symmetry of the associated operator $\Psi$. It is known that the gradient of scalar curvature $\tau$ satisfies (cf. [22])

$$\frac{1}{2}\nabla \tau =\sum _{\alpha =1}^m\left(D_{F_{\alpha }}T\right)\left(F_{\alpha }\right).$$

(12)

Thus, on using Equation (12) in (11), we arrive at

$$Ric\left(F,\mathbf{u}\right)=Ric\left(F,\nabla \sigma \right)-\tau F\left(\sigma \right)-\frac{1}{2}\sigma F\left(\tau \right)-g\left(F,\sum _{\alpha =1}^m\left({\nabla }_{F_{\alpha }}\Psi \right)\left(F_{\alpha }\right)\right)$$

(13)

and, therefore,

$$T\left(\mathbf{u}\right)=T\left(\nabla \sigma \right)-\tau \nabla \sigma -\frac{1}{2}\rho \nabla \tau -\sum _{\alpha =1}^m\left({\nabla }_{F_{\alpha }}\Psi \right)\left(F_{\alpha }\right).$$

(14)

Lemma 1. 

For a σ-RVF $\mathbf{u}$ on a T-manifold $\left(N^m,g\right)$, the associated operator Ψ satisfies

$$\left(D_E\Psi \right)\left(F\right)=R\left(E,\mathbf{u}\right)F-Ric\left(E,F\right)\nabla \sigma +F\left(\sigma \right)T\left(E\right),E,F\in \Gamma \left(N^m\right).$$

Proof. 

Suppose that $\mathbf{u}$ is a $\sigma$-RVF on a T-manifold $\left(N^m,g\right)$. Then, Equation (10) changes to

$$\left(D_E\Psi \right)\left(F\right)-\left(D_F\Psi \right)\left(E\right)=R\left(E,F\right)\mathbf{u}-E\left(\sigma \right)T\left(F\right)+F\left(\sigma \right)T\left(E\right).$$

(15)

Now, using the fact that the 2-form $d\beta$ in Equation (8) is closed and the associated operator $\Psi$ is skew symmetric, we have

$$g\left(\left(D_E\Psi \right)\left(F\right)-\left(D_F\Psi \right)\left(E\right),G\right)+g\left(\left(D_G\Psi \right)\left(E\right),F\right)=0$$

and employing Equation (15) in the above equation yields

$$g\left(R\left(E,F\right)\mathbf{u}-E\left(\sigma \right)T\left(F\right)+F\left(\sigma \right)T\left(E\right),G\right)+g\left(\left(D_G\Psi \right)\left(E\right),F\right)=0.$$

Thus, we have

$$g\left(\left(D_G\Psi \right)\left(E\right),F\right)=g\left(R\left(G,\mathbf{u}\right)E,F\right)+E\left(\sigma \right)g\left(T\left(G\right),F\right)-Ric\left(E,G\right)g\left(\nabla \sigma ,F\right)$$

and this proves the lemma. □

On a Riemannian manifold $\left(N^m,g\right)$ possessing a $\sigma$-RVF $\mathbf{u}$, we have the second-order differential operator ${\nabla }^2\mathbf{u}$ defined by

$$\left({\nabla }^2\mathbf{u}\right)\left(E,F\right)=D_E{D}_F\mathbf{u}-D_{D_{E}F}\mathbf{u},E,F\in \Gamma \left(N^m\right)$$

and its trace

$$\Delta \mathbf{u}=\sum _{\alpha =1}^m\left({\nabla }^2\mathbf{u}\right)\left(E_{\alpha },E_{\alpha }\right)$$

(16)

is the rough Laplacian of the $\sigma$-RVF $\mathbf{u}$.

Lemma 2. 

On a connected T-manifold $\left(N^m,g\right)$, the scalar curvature τ is a constant, and for a σ-RVF $\mathbf{u}$ on a connected T-manifold $\left(N^m,g\right)$ with associated operator Ψ, the rough Laplacian satisfies

$$\Delta \mathbf{u}=T\left(\nabla \sigma \right)+\sum _{\alpha =1}^m\left(D_{E_{\alpha }}\Psi \right)\left(E_{\alpha }\right),$$

where $\left\{E_1,\dots ,E_m\right\}$ is a local frame on $N^m$.

Proof. 

First, note that for a T-manifold $\left(N^m,g\right)$, using Equation (3), we have

$$\begin{array}{cc}\hfill E\left(\tau \right)& =E\sum _{\alpha =1}^{m}g\left(T\left(E_{\alpha }\right),E_{\alpha }\right)\hfill \\ \hfill & =\sum _{\alpha =1}^{m}g\left(\left(D_{E}T\right)\left(E_{\alpha }\right)+T\left(D_E{E}_{\alpha }\right),E_{\alpha }\right)+\sum _{\alpha =1}^{m}g\left(T\left(E_{\alpha }\right),D_E{E}_{\alpha }\right)\hfill \\ \hfill & =\sum _{\alpha =1}^{m}g\left(\left(D_{E_{\alpha }}T\right)\left(E\right),E_{\alpha }\right)+2\sum _{\alpha =1}^{m}g\left(T\left(E_{\alpha }\right),D_E{E}_{\alpha }\right)\hfill \\ \hfill & =\sum _{\alpha =1}^{m}g\left(E,\left(D_{E_{\alpha }}T\right)\left(E_{\alpha }\right)\right)+2\sum _{\alpha =1}^{m}g\left(T\left(E_{\alpha }\right),D_E{E}_{\alpha }\right).\hfill \end{array}$$

(17)

Note that

$$D_E{E}_{\alpha }=\sum _k{\wedge }_{\alpha }^k\left(E\right)E_k,T\left(E_{\alpha }\right)=\sum _j{\mu }_{\alpha }^j{E}_j,$$

where the connection forms ${\wedge }_{\alpha }^k$ are skew symmetric and coefficients ${\mu }_{\alpha }^j$ are symmetric and, as such, we have

$$\sum _{\alpha =1}^{m}g\left(T\left(E_{\alpha }\right),D_E{E}_{\alpha }\right)=0.$$

Consequently, Equation (17) yields

$$\nabla \tau =\sum _{\alpha =1}^m\left(D_{E_{\alpha }}T\right)\left(E_{\alpha }\right).$$

Combining it with Equation (11), we obtain $\nabla \tau =0$, i.e., the scalar curvature $\tau$ of a T-manifold is a constant.

Employing Equation (9), we have

$$\left({\nabla }^2\mathbf{u}\right)\left(E,F\right)=E\left(\sigma \right)T\left(F\right)+\sigma \left(D_{E}T\right)\left(F\right)+\left(D_E\Psi \right)\left(F\right)$$

and taking the trace in the above equation, while using Equation (11) with $\nabla \tau =0$, we obtain

$$\Delta \mathbf{u}=T\left(\nabla \sigma \right)+\sum _{\alpha =1}^m\left(D_{E_{\alpha }}\Psi \right)\left(E_{\alpha }\right).$$

□

Next, the sphere $S_{\alpha }^m$ of constant curvature $\alpha$ possesses a $\sigma$-RVF induced by a coordinate unit vector field $\frac{\partial }{\partial u}$ on the Euclidean space $R^{m+1}$. Indeed, on treating $S_{\alpha }^m$ as an embedded surface in $R^{m+1}$ with unit normal $\zeta$ and Weingarten operator—$\sqrt{\alpha }I$, we express $\frac{\partial }{\partial u}$ as

$$\frac{\partial }{\partial u}=\mathbf{u}+f\zeta ,f=⟨\frac{\partial }{\partial u},\zeta ⟩,$$

(18)

where $⟨,⟩$ is a Euclidean inner product and $\mathbf{u}\in \Gamma \left(S_{\alpha }^m\right)$. On taking g as the induced metric on $S_{\alpha }^m$ and D as the Riemannian connection with respect to g and differentiating the above equation with respect to the vector field $E\in \Gamma \left(S_{\alpha }^m\right)$, we have

$$D_E\mathbf{u}=-\sqrt{\alpha }fE,\nabla f=\sqrt{\alpha }\mathbf{u}.$$

(19)

Using the first equation in (19), it follows that

$${\pounds }_{\mathbf{u}}g=-2\sqrt{\alpha }fg$$

and for the Ricci tensor of $S_{\alpha }^m$, we have

$$Ric=(m-1)\alpha g,\tau =m(m-1)\alpha .$$

(20)

Hence, the vector field $\mathbf{u}$ on $S_{\alpha }^m$ obeys

$$\frac{1}{2}{\pounds }_{\mathbf{u}}g=\sigma Ric,\sigma =-\frac{1}{(m-1)\sqrt{\alpha }}f,$$

(21)

i.e., $\mathbf{u}$ is a $\sigma$-RVF on $S_{\alpha }^m$.

Moreover, note that Equation (21) in view of Equation (19) confirms

$$\begin{array}{ccc}\hfill Hess\left(\sigma \right)\left(E,F\right)& =& g\left(D_E\nabla \sigma ,F\right)\hfill \\ & =& -\frac{1}{(m-1)\sqrt{\alpha }}g\left(D_E\nabla f,F\right)\hfill \\ & =& -\frac{1}{m-1}g\left(D_E\mathbf{u},F\right)\hfill \\ & =& \frac{\sqrt{\alpha }f}{m=1}g\left(E,F\right),\hfill \end{array}$$

i.e.,

$$Hess\left(\sigma \right)=-\alpha \sigma g,\Delta \sigma =-m\alpha \sigma .$$

(22)

Combining Equations (20) and (22), we see that the function $\sigma$ of the $\sigma$-RVF $\mathbf{u}$ on $S_{\alpha }^m$ satisfies the static fluid equation

$$\sigma Ric-Hess\left(\sigma \right)=\frac{1}{m}\left(\tau \sigma -\Delta \sigma \right)g.$$

(23)

We investigate now whether $\sigma$ is a nontrivial solution. If $\sigma$ was a constant, by virtue of Equation (21), it would mean that f was a constant, and, in turn, by (19), it would mean that $\mathbf{u}=0$ and, by the same equation, would imply $f=0$. Inserting this information in (18), we have $\frac{\partial }{\partial u}=0$, a contradiction. Hence, $\sigma$ is a nontrivial solution of the static fluid equation on $S_{\alpha }^m$.

3. $\mathbf{\sigma }$-Ricci Vector Fields on $\mathbf{T}$-Manifolds

In this section, we consider an m-dimensional T-manifold $(N^m,g)$ that possesses a closed $\sigma$-RVF $\mathbf{u}$. It is interesting to observe that, in this situation, the vector field $\mathbf{u}-\nabla \sigma$ plays an interesting role while treating the Ricci operator T of $(N^m,g)$. Note that, by Lemma 2, the scalar curvature $\tau$ of a T-manifold $(N^m,g)$ is a constant and we put $\tau =m(m-1)\alpha$, for a constant $\alpha$. Here, we prove the following result.

Theorem 1. 

An m-dimensional, $m>1$, complete, and simply connected T-manifold $(N^m,g)$ with positive scalar curvature τ admits a nonzero closed σ-RVF $\mathbf{u}$, $\sigma \ne 0$ satisfying

$$T\left(\mathbf{u}-\nabla \sigma \right)=\frac{\tau }{m}\left(\mathbf{u}-\nabla \sigma \right),$$

if and only if $(N^m,g)$ is isometric to $S_{\alpha }^m$, where $\tau =m(m-1)\alpha$.

Proof. 

Suppose that the complete and simply connected T-manifold $(N^m,g)$, $m>1$, of scalar curvature $\tau >0$, admits a nonzero closed $\sigma$-RVF $\mathbf{u}$, $\sigma \ne 0$, which satisfies

$$T\left(\mathbf{u}-\nabla \sigma \right)=\frac{\tau }{m}\left(\mathbf{u}-\nabla \sigma \right).$$

(24)

As the $\sigma$-RVF $\mathbf{u}$ is closed, its associated operator $\Psi =0$, and by Lemma 2, the scalar curvature $\tau$ is a constant, and Equation (14) becomes

$$T\left(\mathbf{u}\right)=T\left(\nabla \sigma \right)-\tau \nabla \sigma .$$

(25)

Treating it with Equation (24) yields

$$\frac{\tau }{m}\left(\mathbf{u}-\nabla \sigma \right)=-\tau \nabla \sigma$$

and, as $\tau >0$, it transforms into

$$\mathbf{u}=-(m-1)\nabla \sigma .$$

(26)

Note that, by Equation (9), we have $\mathrm{div}\mathbf{u}=\sigma \tau$, and taking the divergence in Equation (26) gives

$$\sigma \tau =-(m-1)\Delta \sigma .$$

(27)

Now, inserting the value of $\nabla \sigma$ from Equation (26) into Equation (25), we arrive at

$$T\left(\mathbf{u}\right)=-\frac{m-1}{m}\tau \nabla \sigma .$$

(28)

Note that as $\mathbf{u}$ is closed, Equation (9) has the form

$$D_E\mathbf{u}=\sigma T\left(E\right),E\in \Gamma \left(N^m\right).$$

(29)

Next, we intend to compute the divergence $\mathrm{div}\left(T\mathbf{u}\right)$ and we proceed by choosing a local frame $\left\{E_1,\dots ,E_m\right\}$ and using Equation (29)

$$\begin{array}{ccc}\hfill \mathrm{div}\left(T\mathbf{u}\right)& =& \sum _{\alpha =1}^{m}g\left({\nabla }_{E_{\alpha }}T\mathbf{u},E_{\alpha }\right)\hfill \\ & =& \sum _{\alpha =1}^{m}g\left(\left({\nabla }_{E_{\alpha }}T\right)\left(\mathbf{u}\right)+T\left({\nabla }_{E_{\alpha }}\mathbf{u}\right),E_{\alpha }\right)\hfill \\ & =& \sum _{\alpha =1}^{m}g\left(\mathbf{u},\left({\nabla }_{E_{\alpha }}T\right)\left(E_{\alpha }\right)\right)+\sum _{\alpha =1}^{m}g\left({\nabla }_{E_{\alpha }}\mathbf{u},T\left(E_{\alpha }\right)\right).\hfill \end{array}$$

Note that on T-manifold $(N^m,g)$, by Lemma 2, $\tau$ is a constant and, thus, employing Equations (12) and (29), we arrive at

$$\mathrm{div}\left(T\mathbf{u}\right)=\sigma {∥T∥}^2.$$

Now, utilizing this equation in Equation (28) yields

$$\sigma {∥T∥}^2=-\frac{m-1}{m}\tau \Delta \sigma .$$

(30)

Inserting Equation (27) in the above equation gives

$$\sigma {∥T∥}^2=\frac{1}{m}\sigma {\tau }^2,$$

i.e.,

$$\sigma \left({∥T∥}^2-\frac{1}{m}{\tau }^2\right)=0.$$

As $N^k$ is connected (being simply connected) and $\sigma \ne 0$, in this situation, the above equation yields

$${∥T∥}^2=\frac{1}{m}{\tau }^2.$$

(31)

However, Equation (31) is the equality in Schwartz’s inequality

$${∥T∥}^2\ge \frac{1}{m}{\tau }^2.$$

Hence, equality (31) holds if and only if

$$T=\frac{\tau }{m}I$$

and Equation (29) changes to

$$D_E\mathbf{u}=\frac{\tau }{m}\rho E,E\in \Gamma \left(N^m\right).$$

Thus, on employing Equation (26) in the above equation, we confirm

$$D_E\nabla \sigma =-\frac{\tau }{m(m-1)}\sigma E,E\in \Gamma \left(N^m\right).$$

(32)

Note that as $\mathbf{u}\ne 0$ by Equation (26), the function $\sigma$ is a non-constant function and, also, $\tau$ being a positive constant, letting $\tau =m(m-1)\alpha$, we obtain a positive constant $\alpha$ and Equation (32) is Obata’s equation

$$Hess\left(\sigma \right)=-\alpha \rho g,$$

proving that $(N^m,g)$ is isometric to the sphere $S_{\alpha }^m$ (cf. [18,21]).

Conversely, suppose that $(N^m,g)$ is isometric to the sphere $S_{\alpha }^m$. Then, by Equations (19)–(21), there is a nonzero $\sigma$-RVF $\mathbf{u}$ on $S_{\alpha }^m$ and, as seen earlier, the function $\sigma \ne 0$ and is a non-constant function. Moreover, the Ricci operator of $S_{\alpha }^m$ being

$$T=\frac{\tau }{m}I,$$

the condition

$$T\left(\mathbf{u}-\nabla \sigma \right)=\frac{\tau }{m}\left(\mathbf{u}-\nabla \sigma \right)$$

holds, and this finishes the proof. □

In an earlier result, we considered a closed $\sigma$-RVF $\mathbf{u}$ on an m-dimensional T-manifold $(N^m,g)$ to find a characterization of the sphere $S_{\alpha }^m$. Next, we consider a $\sigma$-RVF $\mathbf{u}$ on an m-dimensional T-manifold $(N^m,g)$ not necessarily closed and prove the following.

Theorem 2. 

An m-dimensional compact and connected T-manifold $\left(N^m,g\right)$, $m>2$ of positive scalar curvature τ admits a σ-RVF $\mathbf{u}$ with associated operator Ψ, $\sigma \ne 0$, $\Delta \mathbf{u}=-\frac{\tau }{m(m-1)}\mathbf{u}$ and the Ricci curvature $Ric\left(\mathbf{u},\mathbf{u}\right)$ satisfies

$$\underset{N^m}{\int }Ric\left(\mathbf{u},\mathbf{u}\right)\ge \underset{N^m}{\int }\left[\frac{m-1}{m}{\tau }^2{\sigma }^2+{∥\Psi ∥}^2\right]$$

if and only if $\left(N^m,g\right)$ is isometric to $S_{\alpha }^m$, where $\tau =m(m-1)\alpha$.

Proof. 

Let an m-dimensional T-manifold $(N^m,g)$, $m>2$, with scalar curvature $\tau >0$ be equipped with a $\sigma$-RVF $\mathbf{u}$ with $\sigma \ne 0$ and associated operator $\Psi$ such that

$$\Delta \mathbf{u}=-\frac{\tau }{m(m-1)}\mathbf{u}$$

(33)

and

$${\int }_{N^m}Ric\left(\mathbf{u},\mathbf{u}\right)\ge {\int }_{N^m}\left[\frac{m-1}{m}{\tau }^2{\sigma }^2+{∥\Psi ∥}^2\right].$$

(34)

Using Lemma 1, we have

$$R\left(E,\mathbf{u}\right)F=\left(D_E\Psi \right)\left(F\right)+Ric\left(E,F\right)\nabla \sigma -F\left(\sigma \right)T\left(E\right),E,F\in \Gamma \left(N^m\right)$$

Employing a local frame $\left\{E_1,\dots ,E_m\right\}$ in the above equation, we conclude

$$Ric\left(\mathbf{u},F\right)=Ric\left(\nabla \sigma ,F\right)-\tau F\left(\sigma \right)-\sum _{\alpha =1}^{m}g\left(F,\left(D_{E_{\alpha }}\Psi \right)\left(E_{\alpha }\right)\right),F\in \Gamma \left(N^m\right)$$

and the above equation implies

$$Ric\left(\mathbf{u},\mathbf{u}\right)=Ric\left(\nabla \sigma ,\mathbf{u}\right)-\tau \mathbf{u}\left(\sigma \right)-\sum _{\alpha =1}^{m}g\left(\mathbf{u},\left(D_{E_{\alpha }}\Psi \right)\left(E_{\alpha }\right)\right).$$

Note that, by Equation (9), we have

$$\mathrm{div}\mathbf{u}=\tau \sigma$$

and using

$$\mathrm{div}\left(\sigma \mathbf{u}\right)=\mathbf{u}\left(\sigma \right)+\tau {\sigma }^2,$$

in the above equation containing the expression of $Ric\left(\mathbf{u},\mathbf{u}\right)$, we derive

$$Ric\left(\mathbf{u},\mathbf{u}\right)=Ric\left(\nabla \sigma ,\mathbf{u}\right)+{\tau }^2{\sigma }^2-\tau \mathrm{div}\left(\sigma \mathbf{u}\right)-\sum _{\alpha =1}^{m}g\left(\mathbf{u},\left(D_{E_{\alpha }}\Psi \right)\left(E_{\alpha }\right)\right).$$

(35)

Next, using a local frame $\left\{E_1,\dots ,E_m\right\}$ on $(N^m,g)$, to compute the $\mathrm{div}\left(\Psi \mathbf{u}\right)$, we have, on using the skew symmetry of the associated operator $\Psi$ and Equation (9),

$$\begin{array}{cc}\hfill \mathrm{div}\left(\Psi \mathbf{u}\right)& =\sum _{\alpha =1}^{m}g\left(D_{E_{\alpha }}\Psi \mathbf{u},E_{\alpha }\right)\hfill \\ \hfill & =\sum _{\alpha =1}^{m}g\left(\left(D_{E_{\alpha }}\Psi \right)\left(\mathbf{u}\right)+\Psi \left(\sigma T{E}_{\alpha }+\Psi E_{\alpha }\right),E_{\alpha }\right)\hfill \\ \hfill & =-\sum _{\alpha =1}^{m}g\left(\mathbf{u},\left(D_{E_{\alpha }}\Psi \right)\left(E_{\alpha }\right)\right)-\sigma \sum _{\alpha =1}^{m}g\left(T{E}_{\alpha },\Psi E_{\alpha }\right)-{∥\Psi ∥}^2\hfill \end{array}$$

(36)

Since T is symmetric and the associated operator $\Psi$ is skew symmetric, it follows that

$$\sum _{\alpha =1}^{m}g\left(T{E}_{\alpha },\Psi E_{\alpha }\right)=0$$

(37)

and Equation (36) now becomes

$$\mathrm{div}\left(\Psi \mathbf{u}\right)=-\sum _{\alpha =1}^{m}g\left(\mathbf{u},\left(D_{E_{\alpha }}\Psi \right)\left(E_{\alpha }\right)\right)-{∥\Psi ∥}^2$$

and, inserting this equation into Equation (35), we arrive at

$$Ric\left(\mathbf{u},\mathbf{u}\right)=Ric\left(\nabla \sigma ,\mathbf{u}\right)+{\tau }^2{\sigma }^2-\tau \mathrm{div}\left(\sigma \mathbf{u}\right)+{∥\Psi ∥}^2+\mathrm{div}\left(\Psi \mathbf{u}\right).$$

Note that on a T-manifold $(N^m,g)$, $\tau$ is a constant and keeping this in mind and integrating the above equation brings us to

$${\int }_{N^m}\left(Ric\left(\mathbf{u},\mathbf{u}\right)-Ric\left(\nabla \sigma ,\mathbf{u}\right)-{\tau }^2{\sigma }^2-{∥\Psi ∥}^2\right)=0.$$

(38)

Observe that, by virtue of the symmetry of the operator T and Equations (9), (12) and (37), and the fact that $\tau$ is a constant, we have

$$\begin{array}{cc}\hfill \mathrm{div}\left(T\mathbf{u}\right)& =\sum _{\alpha =1}^{m}g\left(D_{E_{\alpha }}T\mathbf{u},E_{\alpha }\right)\hfill \\ \hfill & =\sum _{\alpha =1}^{m}g\left(\left(D_{E_{\alpha }}T\right)\left(\mathbf{u}\right)+T\left(\sigma T{E}_{\alpha }+\Psi E_{\alpha }\right),E_{\alpha }\right)\hfill \\ \hfill & =\sigma {∥T∥}^2.\hfill \end{array}$$

(39)

Now, using the fact that

$$\mathrm{div}\left(\sigma T\mathbf{u}\right)=Ric\left(\nabla \sigma ,\mathbf{u}\right)+\sigma \mathrm{div}\left(T\mathbf{u}\right)$$

in Equation (39), we arrive at

$$Ric\left(\nabla \sigma ,\mathbf{u}\right)=\mathrm{div}\left(\sigma T\mathbf{u}\right)-{\sigma }^2{∥T∥}^2.$$

Inserting the above equation in Equation (38), we confirm

$${\int }_{N^m}\left(Ric\left(\mathbf{u},\mathbf{u}\right)+{\sigma }^2{∥T∥}^2-{\tau }^2{\sigma }^2-{∥\Psi ∥}^2\right)=0$$

and the above integral could be rearranged as

$${\int }_{N^m}{\sigma }^2\left({∥T∥}^2-\frac{1}{m}{\tau }^2\right)={\int }_{N^m}\left(\frac{m-1}{m}{\tau }^2{\sigma }^2+{∥\Psi ∥}^2\right)-{\int }_{N^m}Ric\left(\mathbf{u},\mathbf{u}\right).$$

(40)

Treating the above equation with the inequality (34), we arrive at

$${\int }_{N^m}{\sigma }^2\left({∥T∥}^2-\frac{1}{m}{\tau }^2\right)\le 0.$$

The integrand in the above inequality by virtue of Schwartz’s inequality is non-negative, and, therefore, we conclude

$${\sigma }^2\left({∥T∥}^2-\frac{1}{m}{\tau }^2\right)=0.$$

As $\sigma \ne 0$ and $N^m$ is connected, we conclude that

$${∥T∥}^2=\frac{1}{m}{\tau }^2,$$

which, being the equality in Schwartz’s inequality, it holds if and only if

$$T=\frac{\tau }{m}I.$$

(41)

Consequently, as $\tau$ is a constant, Equations (14) and (41) combine to arrive at

$$\frac{\tau }{m}\mathbf{u}=\frac{\tau }{m}\left(\nabla \sigma \right)-\tau \nabla \sigma -\sum _{\alpha =1}^m\left({\nabla }_{F_{\alpha }}\Psi \right)\left(F_{\alpha }\right),$$

for a local frame $\left\{E_1,\dots ,E_m\right\}$ on $(N^m,g)$, i.e., we have

$$\frac{\tau }{m}\mathbf{u}=-\frac{m-1}{m}\tau \nabla \sigma -\sum _{\alpha =1}^m\left({\nabla }_{F_{\alpha }}\Psi \right)\left(F_{\alpha }\right).$$

(42)

Moreover, using Equations (33) and (41) with Lemma 2, we obtain the following:

$$-\frac{\tau }{m(m-1)}\mathbf{u}=\frac{\tau }{m}\left(\nabla \sigma \right)+\sum _{\alpha =1}^m\left({\nabla }_{F_{\alpha }}\Psi \right)\left(F_{\alpha }\right).$$

(43)

Adding Equations (42) and (43), we find

$$\frac{m-2}{m\left(m-1\right)}\tau \mathbf{u}=-\frac{m-2}{m}\tau \nabla \sigma$$

and, as $m>2$, $\tau >0$, it confirms

$$\mathbf{u}=-(m-1)\nabla \sigma .$$

Differentiating the above equation and using Equations (9) and (41), we have

$$D_E\nabla \sigma =-\frac{1}{m-1}\left(\frac{\tau }{m}\sigma E+\Psi \left(E\right)\right),E\in \Gamma \left(N^m\right),$$

which, on taking the inner product with E and noticing that $\Psi$ is a skew symmetric operator, leads to

$$Hess\left(\sigma \right)\left(E,E\right)=-\alpha \sigma g\left(E,E\right),E\in \Gamma \left(N^m\right),$$

where $\tau =m(m-1)\alpha$, i.e., $\alpha$ is a positive constant. Now, polarizing the above equation confirms

$$Hess\left(\sigma \right)=-\alpha \sigma g.$$

Hence, $\left(N^m,g\right)$ is isometric to $S_{\alpha }^m$ (cf. [18,21]).

Conversely, suppose that $\left(N^m,g\right)$ is isometric to $S_{\alpha }^m$. Then, by Equation (21), there is a nonzero $\sigma$-RVF $\mathbf{u}$ on $S_{\alpha }^m$ with $\sigma \ne 0$ and, as $\mathbf{u}$ is is closed, the associated operator $\Psi =0$. Moreover, it is obvious that $S_{\alpha }^m$ is a T-manifold. Thus, using Equation (19), we have

$$\begin{array}{ccc}\hfill \left({\nabla }^2\mathbf{u}\right)\left(E,F\right)& =& D_E{D}_F\mathbf{u}-D_{D_{E}F}\mathbf{u}\hfill \\ & =& -\sqrt{\alpha }E\left(f\right)F\hfill \end{array}$$

and, therefore, by treating the above equation with (16), we have

$$\begin{array}{ccc}\hfill \Delta \mathbf{u}& =& \sum _{\alpha =1}^m\left({\nabla }^2\mathbf{u}\right)\left(E_{\alpha },E_{\alpha }\right)\hfill \\ & =& -\sqrt{\alpha }\nabla f,\hfill \end{array}$$

which, by virtue of Equation (19), implies

$$\begin{array}{ccc}\hfill \Delta \mathbf{u}& =& -\alpha \mathbf{u}\hfill \\ & =& -\frac{\tau }{m(m-1)}\mathbf{u},\hfill \end{array}$$

where $\tau =m(m-1)\alpha$. Finally, using Equations (19) and (21), we have

$$\begin{array}{ccc}\hfill \nabla \sigma & =& -\frac{1}{(m-1)\sqrt{\alpha }}\nabla f\hfill \\ & =& -\frac{1}{m-1}\mathbf{u},\hfill \end{array}$$

i.e.,

$${∥\mathbf{u}∥}^2={(m-1)}^2{∥\nabla \sigma ∥}^2.$$

(44)

Now, Equation (22) implies

$$\sigma \Delta \sigma =-m\alpha {\sigma }^2,$$

which, on integrating by parts, confirms

$$\begin{array}{ccc}\hfill {\int }_{S_{\alpha }^m}{∥\nabla \sigma ∥}^2& =& m\alpha {\int }_{S_{\alpha }^m}{\sigma }^2\hfill \\ & =& \frac{1}{m{(m-1)}^2\alpha }{\int }_{S_{\alpha }^m}{\tau }^2{\sigma }^2.\hfill \end{array}$$

(45)

The Ricci curvature of $S_{\alpha }^m$ is

$$Ric\left(\mathbf{u},\mathbf{u}\right)=(m-1)\alpha {∥\mathbf{u}∥}^2$$

and, thus, using $\Psi =0$ and Equations (44) and (45), we conclude

$$\begin{array}{cc}\hfill {\int }_{S_{\alpha }^m}Ric\left(\mathbf{u},\mathbf{u}\right)& ={\int }_{S_{\alpha }^m}(m-1)\alpha {∥\mathbf{u}∥}^2\hfill \\ & ={\int }_{S_{\alpha }^m}{(m-1)}^3\alpha {∥\nabla \sigma ∥}^2\hfill \\ \hfill & ={\int }_{S_{\alpha }^m}\left[\frac{m-1}{m}{\tau }^2{\sigma }^2+{∥\Psi ∥}^2\right]\hfill \end{array}$$

(46)

and this completes the proof. □

4. $\mathbf{\sigma }$-Ricci Vector Fields on Static Spaces

Now, we are interested in a $\sigma$-RVF $\mathbf{u}$, not necessarily closed, on a Riemannian manifold $\left(N^m,g\right)$ with function $\sigma$ as a nontrivial solution of the static fluid Equation (5). Indeed, we prove the following.

Theorem 3. 

If an m-dimensional compact and connected Riemannian manifold $\left(N^m,g\right)$ admits a σ-RVF $\mathbf{u}$ with associated operator Ψ, such that σ is a nontrivial solution of the static perfect fluid equation, for a positive constant α and the Ricci curvature $Ric\left(\mathbf{u},\mathbf{u}\right)$, it satisfies

$${\int }_{N^m}Ric\left(\mathbf{u},\mathbf{u}\right)\ge {\int }_{N^m}\left[\frac{m-1}{m}{\tau }^2{\sigma }^2+\frac{1}{m}{\left(\Delta \sigma +m\alpha \sigma \right)}^2+{∥\Psi ∥}^2\right],$$

and $\left(N^m,g\right)$ is isometric to $S_{\alpha }^m$, and the converse also holds.

Proof. 

Assume that $\left(N^m,g\right)$ admits a $\sigma$-RVF $\mathbf{u}$ with associated operator $\Psi$, such that $\sigma$ is a nontrivial solution of the static perfect fluid Equation (5) and the Ricci curvature $Ric\left(\mathbf{u},\mathbf{u}\right)$ satisfies

$${\int }_{N^m}Ric\left(\mathbf{u},\mathbf{u}\right)\ge {\int }_{N^m}\left[\frac{m-1}{m}{\tau }^2{\sigma }^2+\frac{1}{m}{\left(\Delta \sigma +m\alpha \sigma \right)}^2+{∥\Psi ∥}^2\right].$$

(47)

Then, the Hessian operator $H_{\sigma }$ of the function $\sigma$ defined by

$$g\left(H_{\sigma }\left(E\right),F\right)=Hess\left(\sigma \right)\left(E,F\right),$$

by virtue of Equation (5) satisfies

$$H_{\sigma }\left(E\right)=\sigma T\left(E\right)+\frac{1}{m}\left(\Delta \sigma -\tau \sigma \right)E,E\in \Gamma \left(N^m\right)$$

Utilizing Equation (9) in the above equation, we arrive at

$$H_{\sigma }\left(E\right)=D_E\mathbf{u}-\Psi \left(E\right)+\frac{1}{m}\left(\Delta \sigma -\tau \sigma \right)E$$

and, for a positive constant $\alpha$, the above equation could be rearranged as

$$\left(H_{\sigma }+\alpha \sigma I\right)\left(E\right)=D_E\mathbf{u}-\Psi \left(E\right)+\frac{1}{m}\left(\Delta \sigma +m\alpha \sigma -\tau \sigma \right)E,E\in \Gamma \left(N^m\right).$$

Choosing a local frame $\left\{E_1,\dots ,E_m\right\}$, and using the above equation, we compute

$$\begin{array}{ccc}\hfill {∥H_{\sigma }+\alpha \sigma I∥}^2& =& \sum _{j=1}^{m}g\left(\left(H_{\sigma }+\alpha \sigma I\right)\left(E_j\right),\left(H_{\sigma }+\alpha \sigma I\right)\left(E_j\right)\right)\hfill \\ & =& \sum _{j=1}^{m}g\left(D_{E_j}\mathbf{u}-\Psi \left(E_j\right)+\frac{1}{m}\left(\Delta \sigma +m\alpha \sigma -\tau \sigma \right)E_j,\right.\hfill \\ & & \left.D_{E_j}\mathbf{u}-\Psi \left(E_j\right)+\frac{1}{m}\left(\Delta \sigma +m\alpha \sigma -\tau \sigma \right)E_j\right)\hfill \\ & =& {∥D\mathbf{u}∥}^2+{∥\Psi ∥}^2+\frac{1}{m}{\left(\Delta \sigma +m\alpha \sigma -\tau \sigma \right)}^2\hfill \\ & & -2\sum _{j=1}^{m}g\left(D_{E_j}\mathbf{u},\Psi \left(E_j\right)\right)\hfill \\ & & +\frac{2}{m}\left(\Delta \sigma +m\alpha \sigma -\tau \sigma \right)\mathrm{div}\left(\mathbf{u}\right).\hfill \end{array}$$

Now, using Equation (9) and $\mathrm{div}\left(\mathbf{u}\right)=\tau \sigma$ in the above equation, we arrive at

$$\begin{array}{ccc}\hfill {∥H_{\sigma }+\alpha \sigma I∥}^2& =& {∥D\mathbf{u}∥}^2-{∥\Psi ∥}^2+\frac{1}{m}{\left(\Delta \sigma +m\alpha \sigma -\tau \sigma \right)}^2\hfill \\ & & +\frac{2\tau \sigma }{m}\left(\Delta \sigma +m\alpha \sigma -\tau \sigma \right),\hfill \end{array}$$

i.e.,

$${∥H_{\sigma }+\alpha \sigma I∥}^2={∥D\mathbf{u}∥}^2-{∥\Psi ∥}^2+\frac{1}{m}\left(\Delta \sigma +m\alpha \sigma -\tau \sigma \right)\left(\Delta \sigma +m\alpha \sigma +\tau \sigma \right)$$

or

$${∥H_{\sigma }+\alpha \sigma I∥}^2={∥D\mathbf{u}∥}^2-{∥\Psi ∥}^2+\frac{1}{m}{\left(\Delta \sigma +m\alpha \sigma \right)}^2-\frac{1}{m}{\left(\tau \sigma \right)}^2.$$

(48)

We recall the integral formula (cf. [23])

$${\int }_{N^m}{∥D\mathbf{u}∥}^2={\int }_{N^m}\left(Ric\left(\mathbf{u},\mathbf{u}\right)+\frac{1}{2}{\left|{\pounds }_{\mathbf{u}}g\right|}^2-{\left(\mathrm{div}\mathbf{u}\right)}^2\right).$$

Using $\mathrm{div}\left(\mathbf{u}\right)=\tau \sigma$ and the outcome of Equation (1) in the form

$$\frac{1}{2}{\left|{\pounds }_{\mathbf{u}}g\right|}^2=2{\sigma }^2{∥T∥}^2$$

in the above integral equation, we have

$${\int }_{N^m}{∥D\mathbf{u}∥}^2={\int }_{N^m}\left(Ric\left(\mathbf{u},\mathbf{u}\right)+2{\sigma }^2{∥T∥}^2-{\left(\tau \sigma \right)}^2\right).$$

Now, integrating Equation (48) and using the above equation, we arrive at

$$\begin{array}{cc}\hfill {\int }_{N^m}{∥H_{\sigma }+\alpha \sigma I∥}^2& ={\int }_{N^m}\left(Ric\left(\mathbf{u},\mathbf{u}\right)+2{\sigma }^2{∥T∥}^2-\frac{m+1}{m}{\left(\tau \sigma \right)}^2\right.\hfill \\ \hfill & \left.-{∥\Psi ∥}^2+\frac{1}{m}{\left(\Delta \sigma +m\alpha \sigma \right)}^2\right)\hfill \end{array}$$

(49)

Notice that

$$2{\sigma }^2{∥T∥}^2-\frac{m+1}{m}{\left(\tau \sigma \right)}^2=2{\sigma }^2\left({∥T∥}^2-\frac{1}{m}{\tau }^2\right)-\frac{m-1}{m}{\left(\tau \sigma \right)}^2$$

(50)

and, by Equation (5), we have that

$$\sigma T\left(E\right)-\frac{1}{m}\tau \sigma E=H_{\sigma }\left(E\right)-\frac{1}{m}\left(\Delta \sigma \right)E,$$

which implies

$${\sigma }^2{∥T-\frac{\tau }{m}I∥}^2={∥H_{\sigma }-\frac{1}{m}\left(\Delta \sigma \right)I∥}^2.$$

Combining it with Equation (50), we arrive at

$$2{\sigma }^2{∥T∥}^2-\frac{m+1}{m}{\left(\tau \sigma \right)}^2=2{∥H_{\sigma }-\frac{1}{m}\left(\Delta \sigma \right)I∥}^2-\frac{m-1}{m}{\left(\tau \sigma \right)}^2.$$

(51)

Moreover, we have

$$\begin{array}{cc}\hfill {∥H_{\sigma }-\frac{1}{m}\left(\Delta \sigma \right)I∥}^2& ={∥H_{\sigma }∥}^2+\frac{1}{m}{\left(\Delta \sigma \right)}^2-\frac{2}{m}\left(\Delta \sigma \right)\sum _{j=1}^{m}g\left(H_{\sigma }\left(E_j\right),E_j\right)\hfill \\ \hfill & ={∥H_{\sigma }∥}^2-\frac{1}{m}{\left(\Delta \sigma \right)}^2\hfill \end{array}$$

(52)

Similarly, we have

$${∥H_{\sigma }+\alpha \sigma I∥}^2={∥H_{\sigma }∥}^2+2\alpha \sigma \Delta \sigma +m{\alpha }^2{\sigma }^2$$

and utilizing it in Equation (52), we obtain

$${∥H_{\sigma }-\frac{1}{m}\left(\Delta \sigma \right)I∥}^2={∥H_{\sigma }+\alpha \sigma I∥}^2-2\alpha \sigma \Delta \sigma -m{\alpha }^2{\sigma }^2-\frac{1}{m}{\left(\Delta \sigma \right)}^2,$$

i.e.,

$${∥H_{\sigma }-\frac{1}{m}\left(\Delta \sigma \right)I∥}^2={∥H_{\sigma }+\alpha \sigma I∥}^2-\frac{1}{m}{\left(\Delta \sigma +m\alpha \sigma \right)}^2.$$

Thus, in view of the above equation, (51) assumes the form

$$2{\sigma }^2{∥T∥}^2-\frac{m+1}{m}{\left(\tau \sigma \right)}^2=2{∥H_{\sigma }+\alpha \sigma I∥}^2-\frac{2}{m}{\left(\Delta \sigma +m\alpha \sigma \right)}^2-\frac{m-1}{m}{\left(\tau \sigma \right)}^2.$$

Now, inserting this value in Equation (49), we arrive at

$$\begin{array}{cc}\hfill {\int }_{N^m}{∥H_{\sigma }+\alpha \sigma I∥}^2& ={\int }_{N^m}\left(Ric\left(\mathbf{u},\mathbf{u}\right)+2{∥H_{\sigma }+\alpha \sigma I∥}^2-\frac{m-1}{m}{\left(\tau \sigma \right)}^2\right.\hfill \\ \hfill & \left.-{∥\Psi ∥}^2-\frac{1}{m}{\left(\Delta \sigma +m\alpha \sigma \right)}^2\right),\hfill \end{array}$$

i.e.,

$$\underset{N^m}{\int }{∥H_{\sigma }+\alpha \sigma I∥}^2=\underset{N^m}{\int }\left(\frac{m-1}{m}{\left(\tau \sigma \right)}^2+\frac{1}{m}{\left(\Delta \sigma +m\alpha \sigma \right)}^2+{∥\Psi ∥}^2\right)-\underset{N^m}{\int }Ric\left(\mathbf{u},\mathbf{u}\right).$$

Using inequality (47) in the above equation, we conclude

$$\underset{N^m}{\int }{∥H_{\sigma }+\alpha \sigma I∥}^2\le 0,$$

which proves

$$Hess\left(\sigma \right)=-\alpha \sigma g,$$

where $\alpha >0$ is a constant and $\sigma$, being a nontrivial solution of a static perfect fluid, is a non-constant function. Hence, $\left(N^m,g\right)$ is isometric to $S_{\alpha }^m$ (cf. [18,21]).

The converse is trivial, because, by Equation (21), $S_{\alpha }^m$ admits a $\sigma$-RVF $\mathbf{u}$, and by Equation (23) and the paragraph that follows (23), $\sigma$ is a nontrivial solution of the static perfect fluid equation. Moreover, we have, by Equation (22), that

$$\Delta \sigma +m\alpha \sigma =0$$

and, by Equation (46), we have

$${\int }_{N^m}Ric\left(\mathbf{u},\mathbf{u}\right)={\int }_{N^m}\left[\frac{m-1}{m}{\tau }^2{\sigma }^2+\frac{1}{m}{\left(\Delta \sigma +m\alpha \sigma \right)}^2+{∥\Psi ∥}^2\right]$$

This finishes the proof. □