Trace of Fischer–Marsden Equation on a Riemannian Space

Abstract

Among the important differential equations on a Riemannian space $\left(M,g\right)$ of dimension n are the static perfect fluid equation (SPFE), namely $fRic-Hess\left(f\right)={\displaystyle \frac{1}{n}}\left(f\tau -\Delta f\right)g$, and the Fischer–Marsden equation (FME), namely $\left(\Delta f\right)g+fRic=Hess\left(f\right)$, where $Ric$ is the Ricci tensor, $\tau$ is the scalar curvature of $\left(M,g\right)$ and $Hess\left(f\right)$ and $\Delta f$ are the Hessian and the Laplacian of the smooth function f. The trace of the FME is $\Delta f=-{\displaystyle \frac{\tau }{n-1}}f$, which we call the TFME, and if we combine the TFME with the SPFE, we observe that it reduces to the FME. Thus, in the presence of the TFME on the Riemannian space $\left(M,g\right)$ the fundamental differential equations SPFE and FME are equivalent. In this article, we consider the presence of the TFME on a Riemannian space $\left(M,g\right)$ and study its impact on the Riemannian space $\left(M,g\right)$. The importance of this study follows from the fact that results obtained for Riemannian spaces admitting solutions to the TFME automatically are generalizations of corresponding results on spaces admitting solutions to the FME. First, we show that for a connected and compact Riemannian space $(M,g)$, $dimM=n>1$, with scalar curvature $\tau$ that admits a nontrivial solution f to the TFME, with the Ricci operator Q satisfying $Q\left(\nabla f\right)={\displaystyle \frac{\tau }{n}}\nabla f$, and with the integral of $Ric\left(\nabla f,\nabla f\right)$ having a suitable lower bound, it is necessary and sufficient that $(M,g)$ is an n-sphere ${\mathbf{S}}^n\left(c\right)$. In addition, we show that a compact and connected space $(M,g)$, $dimM=n>1$, admits a nontrivial solution f to the TFME such that the scalar curvature $\tau$ satisfies $(n-1)c<\tau \le n(n-1)c$ for some constant $c>0$, and the Ricci curvature $Ric(\nabla f,\nabla f)$ is bounded below by $(n-1)c$, if and only if $(M,g)$ is an n-sphere ${\mathbf{S}}^n\left(c\right)$. Finally, it is shown that a connected and compact Riemannian space $(M,g)$, $dimM=$ $n>1$, with constant scalar curvature $\tau$ admits a nontrivial solution f to the TFME, with the Ricci operator Q satisfying $Q\left(\nabla f\right)={\displaystyle \frac{\tau }{n}}\nabla f$, if and only if $(M,g)$ is the sphere ${\mathbf{S}}^n\left(c\right)$.

1. Introduction

Differential equations and certain special vector fields have played an immense role in characterizing the geometry of Riemannian spaces (cf. [1,2,3]). Special vector fields, such as conformal, Killing geodesic and concircular vector fields, play an important role in characterizing spheres and Euclidean spaces (see [4,5,6,7,8,9]). Regarding differential equations, in the celebrated work of Obata [10,11], it is shown that a complete and connected Riemannian space $(M,g)$ with $dimM=n$ admits a nontrivial solution to the differential equation

$$Hess\left(f\right)=-cfg,$$

for a positive constant c, and the Hessian $Hess\left(f\right)$ of the function f on M assures that the space $(M,g)$ is the sphere ${\mathbf{S}}^n\left(c\right)$. Equally important is the Fischer–Marsden differential equation (cf. [12])

$$\left(\Delta f\right)g+fRic=Hess\left(f\right),$$

(1)

on a Riemannian space $(M,g)$, where $\Delta f$ is the action of the Laplace operator on the smooth functions f on M, and $Ric$ is the Ricci tensor of g. We denote differential Equation (1) as the FME and call the solution f to (1) a Fischer–Marsden solution. Note that the FME is used in general relativity under the name of the static perfect fluid equation (cf. [13,14,15,16,17,18]), which relates the spacelike sections of static space times. Indeed, by considering a 4-dimensional Ricci-flat Lorentzian space that admits a time-like Killing vector field $\xi$ with integrable orthogonal distribution to it, we obtain the integral submanifold of this distribution that is related to the solution to the FME, with the solution f being the length of $\xi$. In [12], it is observed that the existence of a nontrivial solution to the FME on a connected space ensures that the scalar curvature has to be constant. This observation led Fischer and Marsden to put forward the conjecture in [12], which states that a compact space admitting a nontrivial solution to the FME must be an Einstein space (to read about recent progress, see [19,20]). It is worth noting that, had this conjecture been true, Obata’s result in [10] would have shown that either the space is Ricci flat or it is an n-sphere. An important result by Kobayashi [21] and Lafontaine [22], showing other possibilities, shows that this conjecture of Fischer–Marsden is false. Also, observe that the counterexamples cited in [21,22] are products or warped products. We point out that a similar conjecture known as no-hair conjecture is considered in [23,24].

To understand the importance of the FME, in [25], the authors studied the space $\mathbf{W}$ of the solutions to the FME, and proved $dim\mathbf{W}\le n+1$, which led to the result obtained in [26]. An important outcome of the their work [25] is that the authors showed that any product space ${\mathbf{S}}^m\times N$, where N is Einstein, provides a solution to the FME and thus provides a new counterexample to the conjecture of Fischer–Marsden.

The next important differential equation of a Riemannian space $(M,g)$, $dimM=n$, is the static perfect fluid equation (SPFE) [14]:

$$hRic-Hess\left(h\right)={\displaystyle \frac{1}{n}}\left(h\tau -\Delta h\right)g.$$

(2)

A space $\left(M,g\right)$ which admits a nontrivial solution h to the SPFE (2) is called a static perfect fluid space. The static perfect fluid space has a key role to play in relativity, but it also plays an important role in fluid dynamics and in differential geometry (see [5,8,14,23,27] and the references therein). Observe that the trace of the FME is

$$\Delta f=-{\displaystyle \frac{\tau }{n-1}}f,$$

(3)

and if we substitute this equation into the SPFE, we see that it reduces to the FME. Thus, Equation (3), which, being a trace of the FME, we call the TFME, makes the SPFE and FME equivalent. It is this property of the TFME that makes it an important subject of study among the differential equations on a connected Riemannian space $(M,g)$. It is worth noting that the TFME is the stationary Schrödinger equation (cf. [28,29,30,31]), and the solutions to the stationary Schrödinger equation generalize the harmonic functions, known as L-harmonic functions [32]. It is worth mentioning that the Schrödinger equation has an importance in mechanics, due to its prediction of the behavior of dynamical systems and its use in nonlinear optics, nanomagnetic systems and plasma physics (cf. [31,33,34,35,36]).

It is worth noting that the TFME on a Riemannian space $(M,g)$, $dimM=$ $n>1$, which is

$$\Delta f=-{\displaystyle \frac{\tau }{n-1}}f,$$

on assuming the scalar curvature of $\tau$ is a constant, shows that the nontrivial solution f to TFME is an eigenfunction of the Laplace operator $\Delta$ corresponding to eigenvalue $\tau {\left(n-1\right)}^{-1}$. Thus, a Riemannian space $(M,g)$, $dimM=$ $n>1$, of constant scalar curvature $\tau$ admits a nontrivial solution to the TFME and implies that the first nonzero eigenvalue ${\mu }_1$ of the Laplace operator satisfies

$$\tau \ge \left(n-1\right){\mu }_1>0,$$

and equality holding implies the first nonzero eigenvalue ${\mu }_1=nc$, where $\tau =n(n-1)c>0$, which is similar to the first eigenvalue of the sphere ${\mathbf{S}}^n\left(c\right)$. This motivated us to set the goal of obtaining the conditions under which a compact Riemannian space $(M,g)$, $dimM=$ $n>1$, admitting a nontrivial solution to the TFME is isometric to the sphere ${\mathbf{S}}^n\left(c\right)$.

Thus, having seen the origin of the FME, it becomes interesting to analyze its impact on the geometry and topology of the Riemannian space $\left(M,g\right)$ on which it holds. In this work, we consider a compact Riemannian space $\left(M,g\right)$ admitting a nontrivial solution f to the TFME and the vector $\nabla f$, being an eigenvector of the Ricci operator Q with the eigenvalue ${\displaystyle \frac{\tau }{n}}$, and seek the condition under which $\left(M,g\right)$ is the sphere $S^n\left(c\right)$. Consequently, we see that a proper lower bound on the integral of the Ricci curvature $Ric\left(\nabla f,\nabla f\right)$ yields the desired result.

In the other test conducted in this article, we show that a compact Riemannian space $(M,g)$, $dimM=n>1$, admitting a nontrivial solution f to the TFME, with scalar curvature $\tau$ satisfying

$$n(n-1)c\ge \tau >(n-1)c,$$

where c is a positive constant and Ricci curvature $Ric\left(\nabla f,\nabla f\right)$ has a lower bound $(n-1)c$ is necessary and sufficient for $\left(M,g\right)$ to be the sphere $S^n\left(c\right)$.

In this article, we are studying the impact of the TFME on a compact Riemannian space $(M,g)$, $dimM=$ $n>1$, and our approach to this study is to use analytic tools. However, there are several other approaches to studying the TFME, and one of the most important approaches to studying the TFME on a compact Riemannian space $(M,g)$, $dimM=$ $n>1$

$$\Delta f=-{\displaystyle \frac{\tau }{n-1}}f$$

is through its numerical solution using stability analysis, as suggested in (cf. [37]). In [37], the authors aim to numerically approximate the two-dimensional nonlinear Allen–Cahn equation using the discrete maximum principle and Energy Stability. It would be interesting to conduct a study of the TFME on a compact Riemannian space $(M,g)$, $dimM=$ $n>1$, parallel to that considered in [37]. We thank the anonymous reviewer for pointing out this approach to us.

2. Preliminaries

Let $(M,g)$ be a Riemannian space of dimension n. ∇ denotes the Riemannian connection and $\mathfrak{X}\left(M\right)$ the Lie algebra of smooth vector fields on M. Then the curvature tensor field R of $(M,g)$ is

$$R(Y_1,Y_2)Y_3=\left[{\nabla }_{Y_1},{\nabla }_{Y_2}\right]Y_3-{\nabla }_{[Y_1,Y_2]}{Y}_3,Y_1,Y_2,Y_3\in \mathfrak{X}\left(M\right),$$

(4)

and the Ricci curvature tensor is

$$Ric(Y_1,Y_2)=\sum _{i=1}^{n}g\left(R(w_i,Y_1)Y_2,w_i\right),$$

with $\{w_1,\dots ,w_n\}$ being a local orthonormal frame on M. The Ricci operator Q of $\left(M,g\right)$ is given by

$$Ric(Y_1,Y_2)=g\left(Q{Y}_1,Y_2\right),$$

and the trace of Q gives the scalar curvature $\tau$, that is,

$$\tau =TrQ=\sum _{i=1}^{n}Ric(w_i,w_i).$$

Moreover, the gradient of the scalar curvature has the following expression:

$${\displaystyle \frac{1}{2}}\nabla \tau =\sum _{i=1}^n\left({\nabla }_{w_i}Q\right)\left(w_i\right),$$

(5)

where

$$\left({\nabla }_{Y_1}Q\right)\left(Y_2\right)={\nabla }_{Y_1}Q{Y}_2-Q\left({\nabla }_{Y_1}{Y}_2\right),Y_1,Y_2\in \mathfrak{X}\left(M\right).$$

If h is a smooth function on M, then the Hessian operator $A_h$ is a symmetric operator given by

$$A_{h}Y={\nabla }_Y\nabla h,Y\in \mathfrak{X}\left(M\right),$$

and the Hessian of h, denoted by $Hess\left(h\right)$, is defined as

$$Hess\left(h\right)(Y_1,Y_2)=g(A_h{Y}_1,Y_2),Y_1,Y_2\in \mathfrak{X}\left(M\right).$$

It follows that

$$R\left(Y_1,Y_2\right)\nabla h=\left({\nabla }_{Y_1}{A}_h\right)\left(Y_2\right)-\left({\nabla }_{Y_2}{A}_h\right)\left(Y_1\right),Y_1,Y_2\in \mathfrak{X}\left(M\right).$$

(6)

Note that, on a compact Riemannian space $(M,g)$, we have the following (cf. [9]):

$$\underset{M}{\int }\left(Ric\left(\mathbf{u},\mathbf{u}\right)+{\displaystyle \frac{1}{2}}{∥{\pounds }_{\mathbf{u}}g∥}^2-{∥\nabla \mathbf{u}∥}^2-{\left(\mathrm{div}\mathbf{u}\right)}^2\right)=0,$$

(7)

where ${\pounds }_{\mathbf{u}}g$ stands for the Lie derivative of g in the direction of $\mathbf{u}$ on M. For a function h on M, we observe that

$${\pounds }_{\nabla h}g=2Hess\left(h\right),{∥\nabla \nabla h∥}^2={∥A_h∥}^2,$$

and upon changing $\mathbf{u}$ to $\nabla h$ in Formula (7), we have

$$\underset{M}{\int }{∥A_h∥}^2=\underset{M}{\int }\left({\left(\Delta h\right)}^2-Ric\left(\nabla h,\nabla h\right)\right).$$

(8)

3. TFME on Compact Riemannian Spaces

In this section, we explore the impact of the TFME on a Riemannian space $(M,g)$ and find two interesting consequences. We start with the following:

Theorem 1.

A connected and compact Riemannian space $(M,g)$, $dimM=$ $n>1$, with scalar curvature τ admits a nontrivial solution f to the TFME, where the Ricci operator Q satisfies

$$Q\left(\nabla f\right)={\displaystyle \frac{\tau }{n}}\nabla f,$$

and the Ricci curvature $Ric\left(\nabla f,\nabla f\right)$ satisfies

$$\underset{M}{\int }Ric\left(\nabla f,\nabla f\right)\ge {\displaystyle \frac{1}{n-1}}\underset{M}{\int }\left(g\left(\nabla f,\nabla \left(\tau f\right)\right)-{\displaystyle \frac{1}{n(n-1)}}{\tau }^2{f}^2\right),$$

if and only if $(M,g)$ is the sphere ${\mathbf{S}}^n\left(c\right)$.

Proof. 

Assume that f is a nontrivial solution to the TFME on $\left(M,g\right)$, that is, we have

$$\Delta f=-{\displaystyle \frac{\tau }{n-1}}f.$$

(9)

Now, we intend to compute $\mathrm{div}\left(A_f\nabla f\right)$, and to achieve this, we take a local orthonormal frame $\{w_1,\dots ,w_n\}$:

$$\begin{array}{ccc}\hfill \mathrm{div}\left(A_f\nabla f\right)& =& \sum _{i=1}^{n}g\left({\nabla }_{w_i}{A}_f\nabla f,w_i\right)=\sum _{i=1}^{n}g\left(\left({\nabla }_{w_i}{A}_f\right)\left(\nabla f\right)+A_f^2{w}_i,w_i\right)\hfill \\ & =& \sum _{i=1}^{n}g\left(\left({\nabla }_{w_i}{A}_f\right)\left(\nabla f\right),w_i\right)+\sum _{i=1}^{n}g\left(A_f^2{w}_i,w_i\right).\hfill \end{array}$$

Inserting Equation (6) into the above equation while employing symmetry of the operator $A_f$, we get

$$\begin{array}{ccc}\hfill \mathrm{div}\left(A_f\nabla f\right)& =& \sum _{i=1}^{n}g\left(R\left(w_i,\nabla f\right)\nabla f,w_i\right)+\sum _{i=1}^{n}g\left(\left({\nabla }_{\nabla f}{A}_f\right)\left(w_i\right),w_i\right)+{∥A_f∥}^2\hfill \\ & =& Ric\left(\nabla f,\nabla f\right)+\nabla f\left(Tr{A}_f\right)+{∥A_f∥}^2\hfill \\ & =& Ric\left(\nabla f,\nabla f\right)+\nabla f\left(\Delta f\right)+{∥A_f∥}^2.\hfill \end{array}$$

On integrating the above equation and inserting Equation (9), we conclude that

$$\underset{M}{\int }\left(Ric\left(\nabla f,\nabla f\right)-{\displaystyle \frac{1}{n-1}}g\left(\nabla f,\nabla \left(\tau f\right)\right)+{∥A_f∥}^2\right)=0.$$

This equation can be re-arranged into the following form:

$$\underset{M}{\int }\left({∥A_f∥}^2-{\displaystyle \frac{1}{n}}{\left(\Delta f\right)}^2\right)=\underset{M}{\int }\left({\displaystyle \frac{1}{n-1}}g\left(\nabla f,\nabla \left(\tau f\right)\right)-Ric\left(\nabla f,\nabla f\right)-{\displaystyle \frac{1}{n}}{\left(\Delta f\right)}^2\right).$$

Inserting Equation (9) on the right-hand integrand, we have

$$\underset{M}{\int }\left({∥A_f∥}^2-{\displaystyle \frac{1}{n}}{\left(\Delta f\right)}^2\right)={\displaystyle \frac{1}{n-1}}\underset{M}{\int }\left(g\left(\nabla f,\nabla \left(\tau f\right)\right)-{\displaystyle \frac{1}{n(n-1)}}{\tau }^2{f}^2\right)-\underset{M}{\int }Ric\left(\nabla f,\nabla f\right).$$

Using the condition in the statement of the Theorem leads to

$$\underset{M}{\int }\left({∥A_f∥}^2-{\displaystyle \frac{1}{n}}{\left(\Delta f\right)}^2\right)\le 0.$$

(10)

Note that according to Cauchy–Schwartz inequality, it follows that

$${∥A_f∥}^2\ge {\displaystyle \frac{1}{n}}{\left(\Delta f\right)}^2,$$

and combining it with inequality (10), leads to

$${∥A_f∥}^2={\displaystyle \frac{1}{n}}{\left(\Delta f\right)}^2.$$

The results in

$$A_f={\displaystyle \frac{\Delta f}{n}}I,$$

which leads to

$$A_f=-{\displaystyle \frac{\tau }{n(n-1)}}fI.$$

(11)

From the above equation, we compute

$$\left({\nabla }_{Y_1}{A}_f\right)\left(Y_2\right)=-{\displaystyle \frac{1}{n(n-1)}}{Y}_1\left(\tau f\right)Y_2,$$

and inserting this equation into Equation (6) gives

$$R\left(Y_1,Y_2\right)\nabla f=-{\displaystyle \frac{1}{n(n-1)}}\left(Y_1\left(\tau f\right)Y_2-Y_2\left(\tau f\right)Y_1\right).$$

Tracing the above equation yields

$$Ric\left(Y_2,\nabla f\right)={\displaystyle \frac{1}{n}}{Y}_2\left(\tau f\right),$$

that is,

$$Q\left(\nabla f\right)={\displaystyle \frac{1}{n}}\nabla \left(\tau f\right).$$

Now, using the condition $Q\left(\nabla f\right)={\displaystyle \frac{\tau }{n}}\nabla f$ in the above equation yields

$$f\nabla \tau =0.$$

Since f is a nontrivial solution to the TFME and M is connected, in the above equation, this forces $\tau$ to be a constant. On multiplying by f, the TFME gives

$$f\Delta f=-{\displaystyle \frac{\tau }{(n-1)}}{f}^2,$$

which, on integrating and using the fact that $\tau$ is a constant, gives

$$\underset{M}{\int }{∥\nabla f∥}^2={\displaystyle \frac{\tau }{n-1}}\underset{M}{\int }{f}^2.$$

This proves that $\tau$ is positive, and setting $\tau =n(n-1)c$ for a constant $c>0$, Equation (11) takes the form

$$A_f=-cfI.$$

Hence, $\left(M,g\right)$ is isometric to the sphere ${\mathbf{S}}^n\left(c\right)$ (cf. [10,38]).

Conversely, let $(M,g)$ be isometric to ${\mathbf{S}}^n\left(c\right)$. Then, we know that the first nonzero eigenvalue of ${\mathbf{S}}^n\left(c\right)$ is $nc$, and thus, taking the eigenfunction f corresponding to eigenvalue $nc$, we see that

$$\Delta f=-ncf=-{\displaystyle \frac{\tau }{n-1}}f$$

(12)

holds as $\tau =n(n-1)c$. The above equation gives

$$\underset{{\mathbf{S}}^n\left(c\right)}{\int }{∥\nabla f∥}^2=nc\underset{{\mathbf{S}}^n\left(c\right)}{\int }{f}^2.$$

(13)

Moreover, as $Q={\displaystyle \frac{\tau }{n}}I$, the condition $Q\left(\nabla f\right)={\displaystyle \frac{\tau }{n}}\nabla f$ holds. Finally, we have

$$Ric\left(\nabla f,\nabla f\right)=\left(n-1\right)c{∥\nabla f∥}^2,$$

which, on integration by virtue of Equation (13), yields

$$\underset{{\mathbf{S}}^n\left(c\right)}{\int }Ric\left(\nabla f,\nabla f\right)=n(n-1)c^2\underset{{\mathbf{S}}^n\left(c\right)}{\int }{f}^2.$$

(14)

Now, we use $\tau =n(n-1)c$ and Equation (13) to compute

$${\displaystyle \frac{1}{n-1}}\underset{{\mathbf{S}}^n\left(c\right)}{\int }\left(g\left(\nabla f,\nabla \left(\tau f\right)\right)-{\displaystyle \frac{1}{n(n-1)}}{\tau }^2{f}^2\right)=n(n-1)c^2\underset{{\mathbf{S}}^n\left(c\right)}{\int }{f}^2,$$

and combining it with Equation (14), we see that the remaining condition also holds. □

In our second test, we use the existence of nontrivial solution f to the TFME on a compact Riemannian space $\left(M,g\right)$ and some suitable bounds on the scalar curvature to obtain the following result.

Theorem 2.

Let $(M,g)$ be a compact Riemannian space of dimension n with scalar curvature τ. Then $(M,g)$ admits a nontrivial solution f to the TFME such that the scalar curvature τ satisfies $(n-1)c<\tau \le n(n-1)c$ for a constant $c>0$, and the Ricci curvature $Ric(\nabla f,\nabla f)$ has lower bound $(n-1)c{∥\nabla f∥}^2$, if and only if the Riemannian space $(M,g)$ is the sphere ${\mathbf{S}}^n\left(c\right)$.

Proof. 

Let f be a nontrivial solution to the equation

$$\Delta f=-{\displaystyle \frac{\tau }{n-1}}f,$$

(15)

and let $(n-1)c<\tau \le n(n-1)c$ hold for $c>0$. Then, we get

$$f\Delta f=-{\displaystyle \frac{\tau }{n-1}}{f}^2,$$

which, on integration, gives

$$\underset{M}{\int }{∥\nabla f∥}^2={\displaystyle \frac{1}{n-1}}\underset{M}{\int }\tau f^2.$$

(16)

Now, choosing a local frame $\left\{w_1,\dots ,w_n\right\}$, we have

$$\begin{array}{ccc}\hfill {∥A_f+cfI∥}^2& =& \sum _{i=1}^{n}g\left(A_f{w}_i+cf{w}_i,A_f{w}_i+cf{w}_i\right)\hfill \\ & =& {∥A_f∥}^2+n{c}^2{f}^2+2cf\Delta f.\hfill \end{array}$$

Integrating the above expression while using Equation (15) leads to

$$\underset{M}{\int }{∥A_f+cfI∥}^2=\underset{M}{\int }\left({∥A_f∥}^2+c\left(nc-2{\displaystyle \frac{\tau }{n-1}}\right)f^2\right),$$

and combining it with Equation (8), we conclude that

$$\underset{M}{\int }{∥A_f+cfI∥}^2=\underset{M}{\int }\left({(\Delta f)}^2+c\left(nc-2{\displaystyle \frac{\tau }{n-1}}\right)f^2-Ric\left(\nabla f,\nabla f\right)\right).$$

(17)

Now, using the hypothesis

$$Ric\left(\nabla f,\nabla f\right)\ge (n-1)c{∥\nabla f∥}^2,$$

and inserting Equation (15) into (17), we confirm that

$$\underset{M}{\int }{∥A_f+cfI∥}^2\le \underset{M}{\int }\left(\left({\left({\displaystyle \frac{\tau }{n-1}}\right)}^2+c\left(nc-2{\displaystyle \frac{\tau }{n-1}}\right)\right)f^2-(n-1)c{∥\nabla f∥}^2\right).$$

Treating the above inequality with Equation (16) yields

$$\underset{M}{\int }{∥A_f+cfI∥}^2\le \underset{M}{\int }\left({\displaystyle \frac{\tau }{n-1}}-nc\right)\left({\displaystyle \frac{\tau }{n-1}}-c\right)f^2.$$

Also, the condition $(n-1)c<\tau \le n(n-1)c$ makes the integrand on the right-hand side of the above inequality non-positive; therefore, we obtain

$$\underset{M}{\int }{∥A_f+cfI∥}^2\le 0,$$

and it proves that

$$A_f+cfI=0.$$

Thus, we have

$${\nabla }_U\nabla f=-cfU,U\in \mathfrak{X}\left(M\right),$$

and it is the same as Obata’s differential equation for non-constant function f and real constant $c>0$ (cf. [10,11]). Hence, $(M,g)$ is isometric to the sphere ${\mathbf{S}}^n\left(c\right)$.

The converse is trivial, as on ${\mathbf{S}}^n\left(c\right)$, one has a eigenfunction f corresponding to the first nonzero eigenvalue $nc$, that is, $\Delta$ $f=-nc$ f, which implies that

$$\tau =n(n-1)c,$$

that is,

$$\Delta f=-{\displaystyle \frac{\tau }{n-1}}f.$$

Thus, f is a nontrivial solution to the TFME, and we have $Ric\left(\nabla f,\nabla f\right)=(n-1)c{∥\nabla f∥}^2$. □

4. Examples

We observe that there are Riemannian spaces that admit nontrivial solutions to the TFME, and we discuss some examples below.

(i)

Consider an n-sphere ${\mathbf{S}}^n\left(c\right)$ of constant curvature c with scalar curvature $\tau =n(n-1)c$ and the first nonzero eigenvalue of the Laplace operator $nc$. Choosing a smooth function f as the eigenfunction of the Laplace operator $\Delta$ corresponding to the eigenvalue $nc$, we have

$$\Delta f=-ncf,$$

that is,

$$\Delta f=-{\displaystyle \frac{\tau }{n-1}}f,$$

which is the TFME.

(ii)

Consider a Riemannian space $\left(M,g\right)$ of dimension $n>1$ that admits a nonhomothetic closed conformal vector field $\xi$ with potential function f, and with scalar curvature $\tau$ that is constant along the integral curves of $\xi$. Then, we have

$${\nabla }_Y\xi =fY,Y\in \mathfrak{X}\left(M\right),$$

(18)

and a straight-forward computation yields

$$R\left(Y_1,Y_2\right)\xi =Y_1\left(f\right)Y_2-Y_2\left(f\right)Y_1,Y_1,Y_2\in \mathfrak{X}\left(M\right).$$

Tracing the above equation yields

$$Ric\left(Y_2,\xi \right)=-(n-1)Y_2\left(f\right),$$

that is,

$$Q\left(\xi \right)=-(n-1)\nabla f.$$

(19)

Now, we wish to find $\mathrm{div}Q\left(\xi \right)$, and to achieve this, we choose a local frame $\{w_1,\dots ,w_n\}$ and use Equation (18); we get

$$\begin{array}{ccc}\hfill \mathrm{div}Q\left(\xi \right)& =& \sum _{i=1}^{n}g\left({\nabla }_{w_i}Q\left(\xi \right),w_i\right)=\sum _{i=1}^{n}g\left(\left({\nabla }_{w_i}Q\right)\left(\xi \right)+fQ\left(w_i\right),w_i\right)\hfill \\ & =& \sum _{i=1}^{n}g\left(\left({\nabla }_{w_i}Q\right)\left(\xi \right),w_i\right)+f\tau .\hfill \end{array}$$

As the operator Q is symmetric, on using Equation (5), we get

$$\mathrm{div}Q\left(\xi \right)={\displaystyle \frac{1}{2}}g\left(\xi ,\nabla \tau \right)+f\tau ,$$

and subjecting it to the assumption that $\tau$ is constant along the trajectories of $\xi$, we conclude that

$$\mathrm{div}Q\left(\xi \right)=f\tau .$$

The divergence of Equation (19) together with the above equations implies that

$$\Delta f=-{\displaystyle \frac{\tau }{n-1}}f,$$

which is a TFME.

5. Concluding Remarks

In this article, we have used the nontrivial solution to the TFME,

$$\Delta f=-{\displaystyle \frac{\tau }{n-1}}f,$$

(20)

on a connected and compact Riemannian space $\left(M,g\right)$ in order to find two characterizations for the unit sphere ${\mathbf{S}}^n\left(c\right)$ (cf. Theorems 1 and 2). Moreover, in example (ii), we used a nonhomothetic closed conformal vector field $\xi$ with potential function f on a space $\left(M,g\right)$ with scalar curvature $\tau$ that is constant along the integral curves of $\xi$ to show the existence of the TFME on $\left(M,g\right)$.

• It would be interesting to study the impact on the geometry of a connected Riemannian space $\left(M,g\right)$ admitting a nonhomothetic conformal vector field $\xi$ (not necessarily closed) with potential function f, that is, one that satisfies

  $${\pounds }_{\xi }g=2fg$$

  and the potential function satisfying the TFME (20). In addition to the above, assuming $\left(M,g\right)$ is compact, it would be interesting to find yet another characterization of the sphere.
• We would like to comment on the condition in the statement of Theorem 1, namely

  $$\underset{M}{\int }Ric\left(\nabla f,\nabla f\right)\ge {\displaystyle \frac{1}{n-1}}\underset{M}{\int }\left(g\left(\nabla f,\nabla \left(\tau f\right)\right)-{\displaystyle \frac{1}{n(n-1)}}{\tau }^2{f}^2\right),$$

  (21)

  where f is a nontrivial solution to the TFME, which appears to be artificially imposed without any geometric consequences. However, this condition is consistent with condition $Q\left(\nabla f\right)={\displaystyle \frac{\tau }{n}}\nabla f$. Recall that the purpose of these restrictions is ultimately to achieve the target, that is, the n-sphere ${\mathbf{S}}^n\left(c\right)$. A more general class of Riemannian spaces than ${\mathbf{S}}^n\left(c\right)$, that is, spaces of constant curvature, is the class of constant scalar curvature $\tau$. In this case, Equation (20) implies that f is the eigenfunction of $\Delta$, and consequently, we have

  $$\underset{M}{\int }{∥\nabla f∥}^2={\displaystyle \frac{\tau }{n-1}}\underset{M}{\int }{f}^2.$$
  In view of above equation for constant scalar curvature $\tau$, inequality (21) becomes

  $$\underset{M}{\int }Ric\left(\nabla f,\nabla f\right)\ge {\displaystyle \frac{\tau }{n}}\underset{M}{\int }{∥\nabla f∥}^2,$$

  which is consistent with $Q\left(\nabla f\right)={\displaystyle \frac{\tau }{n}}\nabla f$ (as in this case $Ric\left(\nabla f,\nabla f\right)={\displaystyle \frac{\tau }{n}}{∥\nabla f∥}^2$). Note that in the proof of Theorem 1, constant scalar curvature $\tau$ in combination with $Q\left(\nabla f\right)={\displaystyle \frac{\tau }{n}}\nabla f$, Equation (10) becomes

  $$\underset{M}{\int }\left({∥A_f∥}^2-{\displaystyle \frac{1}{n}}{\left(\Delta f\right)}^2\right)=0,$$

  which leads to the same conclusion. Thus, as a consequence of Theorem 1, we have the following:

Corollary 1.

A connected and compact Riemannian space $(M,g)$, $dimM=$ $n>1$, with constant scalar curvature τ admits a nontrivial solution f to the TFME, with the Ricci operator Q satisfying

$$Q\left(\nabla f\right)={\displaystyle \frac{\tau }{n}}\nabla f,$$

if and only if $(M,g)$ is the sphere ${\mathbf{S}}^n\left(c\right)$.

• It is worth noting that theoretical work and differential equations, such as the FME and TFME considered in this article, are important to general relativity, and ultimately underpin the physics that engineers must model and verify. This indicates the applied significance of fundamental geometric analysis of the FME and TFME in fields like computational physics and engineering. For this type of work, we refer to the work of Guo et al. (cf. [39]).