Ricci Vector Fields

1. Introduction

An m-dimensional complete simply connected Riemannian manifold of constant curvature $\alpha$ is isometric to one of the spaces the m-sphere $S^m\left(\alpha \right)$, the Euclidean space $R^m$ or the hyperbolic space $H^m\left(\alpha \right)$ according as $\alpha >0$, $\alpha =0$ or $\alpha <0$ respectively (cf. [2]). Since, this classification, there has been an interest in obtaining necessary and sufficient conditions on complete Riemannian manifolds so that they are isometric to one of the three model spaces $S^m\left(\alpha \right)$, $R^m$ and $H^m\left(\alpha \right)$ respectively. In that one of most sought questions is in obtaining different characterizations of spheres $S^m\left(\alpha \right)$ among complete Riemannian manifolds. In obtaining these characterizations most of the times conformal and Killing vector fields are used on an m-dimensional complete Riemannian manifold $\left(N^m,g\right)$ (cf. [1,4,5,6,7,8,9,10,11,14,15]). A vector field $\mathbf{u}$ on m-Riemannian manifold $\left(N^m,g\right)$ is a conformal vector field if the Lie derivative ${\pounds }_{\mathbf{u}}g$ has expression

$${\pounds }_{\mathbf{u}}g=2fg,$$

where f is a smooth function called the conformal factor. If $f=0$ in above definition, then $\mathbf{u}$ is called a Killing vector field.

In this paper, we are interested in a vector field $\omega$ on an m-dimensional Riemannian manifold $\left(N^m,g\right)$ that satisfies

$$\frac{1}{2}{\pounds }_{\omega }g=\rho Ric,$$

(1.1)

where ${\pounds }_{\omega }g$ is the Lie-derivative of the metric g with respect to $\omega$, $\rho$ is a smooth function and $Ric$ is the Ricci tensor of $\left(N^m,g\right)$. We call $\omega$ satisfying equation (1.1) a $\rho$-Ricci vector field on $\left(N^m,g\right)$. Naturally, if $\left(N^m,g\right)$ is an Einstein manifold, then a $\rho$-Ricci vector field $\omega$ is a conformal vector field on $\left(N^m,g\right)$ (cf. [4,5,6,7,8,9]). If in the equation (1.1), we take $\rho =0$, then 0-Ricci vector field $\omega$ on $\left(N^m,g\right)$ is a Killing vector field on $\left(N^m,g\right)$ (cf. [10]). A $\rho$-Ricci vector field on $\left(N^m,g\right)$ is also a particular form of potential field of a generalized soliton (cf. [12]), with $\alpha =-\rho$ and $\beta =\gamma =0$.

We could also approach to equation (1.1) in other context (cf. [3]). On the m-dimensional Riemannian manifold $\left(N^m,g\right)$, take a smooth function $\rho$ and consider 1-parameter family of metrics $g\left(t\right)$ satisfying generalized Ricci flow (or $\rho$-Ricci flow) equation

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

(1.2)

To reach a solution of above flow, we take a 1-parameter family of diffeormorphisms ${\phi }_t:N^m\to N^m$ generated by the family of vector fields $\mathbf{W}\left(t\right)$ and $\sigma \left(t\right)$ be a scale factor. Then we are interested in a solution of flow (1.2) of the form

$$g\left(t\right)=\sigma \left(t\right){\phi }_t^{*}\left(g\right).$$

Differentiating above equation with respect to t and substituting $t=0$, while assuming $\sigma \left(0\right)=1$, $\stackrel{.}{\sigma }\left(0\right)=0$, $\mathbf{W}\left(0\right)=\omega$ and using ${\phi }_0=id$, we get

$${\pounds }_{\omega }g-2\rho Ric=0,$$

which is equation (1.1). Thus, a $\rho$-Ricci vector field $\omega$ on $\left(N^m,g\right)$ can be considered as stable solution of the flow (1.2).

We see that as a trivial example on the Euclidean space $R^m$, a constant vector field $\mathbf{a}$ is a $\rho$-Ricci vector field for any smooth function $\rho$ on $R^m$. Similarly on the complex Euclidean space $C^m$ with complex structure J and the vector field

$$\xi =\sum _{i=1}^m{z}^i\frac{\partial }{\partial z^i},$$

where $z^1,..,,z^m$ are Euclidean coordinates, the vector field $\omega =J\xi$ is a $\rho$-Ricci vector field for any smooth function $\rho$ on $C^m$.

Next, we show that on the sphere $S^m\left(\alpha \right)$ of constant curvature $\alpha$, there are many $\rho$-Ricci vector fields. With the imbedding $i:S^m\left(\alpha \right)\to R^{m+1}$ and unit normal $\xi$ and shape operator $-\sqrt{\alpha }I$, on taking a nonzero constant vector field $\mathbf{b}$ on the Euclidean space $R^{m+1}$, we have $\mathbf{b}=\omega +f\xi$, where $f=〈\mathbf{b},\xi 〉$ and $\omega$ is the tangential component of $\mathbf{b}$ to the sphere $S^m\left(\alpha \right)$. Denote the induced metric on the sphere $S^m\left(\alpha \right)$ by g and the Riemannian connection by D. Then differentiating above equation with respect to the vector field X on $S^m\left(\alpha \right)$, we have

$$D_X\omega =-\sqrt{\alpha }fX,\nabla f=\sqrt{\alpha }\omega ,$$

(1.3)

where $\nabla f$ is the gradient of f. Using the first equation in (1.3), it follows that

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

and the Ricci tensor of the sphere $S^m\left(\alpha \right)$ is given by

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

Thus, we see that the vector field $\omega$ on the sphere $S^m\left(\alpha \right)$ satisfies

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

(1.4)

that is, $\omega$ is a $\rho$-Ricci vector field on the sphere $S^m\left(\alpha \right)$. Indeed. for each nonzero constant vector field on the Euclidean space $R^{m+1}$, there is a $\rho$-Ricci vector field on the sphere $S^m\left(\alpha \right)$.

Above example naturally leads to a question: Under what conditions a compact and connected m-dimensional Riemannian manifold $\left(N^m,g\right)$ admitting a $\rho$-Ricci vector field $\omega$ is isometric to a m-sphere $S^m\left(\alpha \right)$?

There are two well known differential equations on a Riemannian manifold $\left(N^m,g\right)$, the first is Obata’s differential equation namely (cf. [14,15]),

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

(1.5)

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

$$Hess\left(\sigma \right)(X,Y)=g\left(D_X\nabla \sigma ,Y\right),$$

for smooth vector fields $X,Y$ on $N^m$. Obata proved that a necessary and sufficient condition for a complete and simply connected Riemannian manifold $\left(N^m,g\right)$ to admit a nontrivial solution of differential equation (1.5) is that $\left(N^m,g\right)$ is isometric to the sphere $S^m\left(\alpha \right)$ (cf. [14,15]). The other differential equation on $\left(N^m,g\right)$ is Fischer-Marsden equation (cf. [13])

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

(1.6)

where $\sigma$ is a smooth function on $N^m$ and $\Delta \sigma =div\left(\nabla \sigma \right)$ is the Laplacian of $\sigma$. We shall use the abbreviation for the above Fischer-Marsden equation as FM-equation. Taking trace in the FM-equation (1.6), we get

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

(1.7)

where $\tau =TrRic$ is the scalar curvature of the Riemannian manifold $\left(N^m,g\right)$. It is known that if $\left(N^m,g\right)$ admits a nontrivial solution of FM-equation, then the scalar curvature $\tau$ is necessarily constant (cf. [13]).

Note that by equation (1.3), the smooth function f on the sphere $S^m\left(\alpha \right)$ has Hessian

$$Hess\left(f\right)\left(X,Y\right)=g\left(D_X\nabla f,Y\right)=\sqrt{\alpha }g\left(D_X\omega ,Y\right)=-\alpha fg\left(X,Y\right),$$

the Laplacian $\Delta f=div\left(\sqrt{\alpha }\omega \right)=-m\alpha f$ and $Ric=(m-1)\alpha g$. Consequently, on $S^m\left(\alpha \right)$, we see that

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

(1.7)

that is, f is a solution of FM-equation on the sphere $S^m\left(\alpha \right)$. If we combine the two, namely a Riemannian manifold $\left(N^m,g\right)$ admits a $\rho$-Ricci vector field $\omega$ such that $\rho$ is a nontrivial solution of the FM-equation on $\left(N^m,g\right)$ and seek additional condition under which $\left(N^m,g\right)$ is isometric to $S^m\left(\alpha \right)$? Notice that the $\rho$-Ricci vector field $\omega$ on the sphere $S^m\left(\alpha \right)$ is a closed vector field. Therefore, in this paper, we use the closed $\rho$-Ricci vector field $\omega$ on a Riemannian manifold $\left(N^m,g\right)$ and answer these two question in section-3, where we find two characterizations of the sphere $S^m\left(\alpha \right)$.

2. Preliminaries

Let $\omega$ be a closed $\rho$-Ricci vector field on an m-dimensional Riemannian manifold $\left(N^m,g\right)$. If $\beta$ is the 1-form dual to $\omega$, that is,

$$\beta \left(X\right)=g\left(\omega ,X\right),X\in \Theta \left(T{N}^m\right),$$

(2.1)

where $\Theta \left(T{N}^m\right)$ is the space of smooth sections of the tangent bundle $T{N}^m$, then we have $d\beta =0$. We denote by ${\nabla }_X$ the covariant derivative operator with respect to the Riemannian connection on $\left(N^m,g\right)$ and notice that for the closed $\rho$-Ricci vector field $\omega$, we have

$$\begin{array}{ccc}\hfill 2g\left({\nabla }_X\omega ,Y\right)& =& g\left({\nabla }_X\omega ,Y\right)+g\left({\nabla }_Y\omega ,X\right)+g\left({\nabla }_X\omega ,Y\right)-g\left({\nabla }_Y\omega ,X\right)\hfill \\ & =& \left({\pounds }_{\omega }g\right)\left(X,Y\right)+d\beta \left(X,Y\right)=2\rho Ric\left(X,Y\right).\hfill \end{array}$$

Thus, for a closed $\rho$-Ricci vector field $\omega$, we have

$${\nabla }_X\omega =\rho TX,X\in \Theta \left(T{N}^m\right),$$

(2.2)

where T is a symmetric operator called Ricci operator given by

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

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

$$R\left(X,Y\right)Z=\left[{\nabla }_X,{\nabla }_Y\right]Z-{\nabla }_{[X.Y]}Z,X,Y,Z\in \Theta \left(T{N}^m\right),$$

and equation (2.2), we get

$$R\left(X,Y\right)\omega =X\left(\rho \right)TY-Y\left(\rho \right)TX+\rho \left(\left({\nabla }_{X}T\right)\left(Y\right)-\left({\nabla }_{Y}T\right)\left(X\right)\right),$$

(2.3)

$X,Y\in \Theta \left(T{N}^m\right)$, where $\left({\nabla }_{X}T\right)\left(Y\right)={\nabla }_{X}TY-T\left({\nabla }_{X}Y\right)$. The scalar curvature $\tau$ of $\left(N^m,g\right)$ is given by $\tau =TrT$, where $TrT$ is the trace of the symmetric operator T. Choosing a local frame $\left\{F_1,..,F_m\right\}$ and using the definition of the Ricci tensor $Ric$

$$Ric\left(X,Y\right)=\sum _{j=1}^{m}g\left(R\left(F_j,X\right)Y,F_j\right),$$

together with equation (1.3), we conclude that

$$Ric\left(Y,\omega \right)=Ric\left(Y,\nabla \rho \right)-\tau Y\left(\rho \right)+\rho g\left(Y,\sum _{j=1}^m\left({\nabla }_{F_j}T\right)\left(F_j\right)\right)-\rho Y\left(\tau \right),$$

(2.4)

where $\nabla \rho$ is the gradient of $\rho$. It is known that the gradient of scalar curvature $\tau$ satisfies (cf. [2])

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

(2.5)

Consequently, equation (2.4) takes the form

$$Ric\left(Y,\omega \right)=Ric\left(Y,\nabla \rho \right)-\tau Y\left(\rho \right)-\frac{1}{2}\rho Y\left(\tau \right)$$

(2.6)

and we have

$$T\left(\omega \right)=T\left(\nabla \rho \right)-\tau \nabla \rho -\frac{1}{2}\rho \nabla \tau .$$

(2.7)

3. Characterizing spheres via $\rho$-Ricci fields

Let $\omega$ be a closed $\rho$-Ricci vector field on an m-dimensional Riemannian manifold $(N^m,g)$. We shall use $\rho$-Ricci vector field and find two characterizations of m-sphere ${\mathbf{S}}^m\left(\alpha \right)$. In our first result, we prove the following result:

Theorem 1.

A closed ρ-Ricci vector field ω on an m-dimensional compact and connected Riemannian manifold $(N^m,g)$, $m>2$ with scalar curvature $\tau \ne 0$ and nonzero nonconstant function ρ satisfies

$${\int }_{M}Ric\left(\omega ,\omega \right)\ge \frac{m-1}{m}{\int }_M{\left(div\omega \right)}^2,$$

if and only if, τ is a positive constant $m(m-1)\alpha$, and $(N^m,g)$ is isometric to $S^m\left(\alpha \right)$.

Proof. 

Let $(N^m,g)$ be an m-dimensional compact and connected Riemannian manifold, $m>2$ with scalar scalar curvature $\tau \ne 0$ and $\omega$ be a closed $\rho$-Ricci vector field defined on $(N^m,g)$ with nonzero and nonconstant function $\rho$ satisfying

$${\int }_{M}Ric\left(\omega ,\omega \right)\ge \frac{m-1}{m}{\int }_M{\left(div\omega \right)}^2.$$

(3.1)

Then using equation (2.2), we have

$$div\omega =\rho \tau .$$

(3.2)

Choosing a local orthonormal frame $\left\{F_1,..,F_m\right\}$ and using

$${∥T∥}^2=\sum _{j=1}^{m}g\left(T{F}_j,T{F}_j\right)$$

and an outcome of equation (2.2) as

$$\left({\pounds }_{\omega }g\right)\left(X,Y\right)=2\rho g\left(TX,Y\right),X,Y\in \Theta \left(T{N}^m\right),$$

we conclude

$$\frac{1}{2}{\left|{\pounds }_{\omega }g\right|}^2=2{\rho }^2{∥T∥}^2.$$

(3.3)

Note that, we have

$$\begin{array}{ccc}\hfill {∥T-\frac{\tau }{m}I∥}^2& =& \sum _{j=1}^{m}g\left(\left(T{E}_j-\frac{\tau }{m}{E}_j\right),\left(T{E}_j-\frac{\tau }{m}{E}_j\right)\right)\hfill \\ & =& {∥T∥}^2+\frac{1}{m}{\tau }^2-2\sum _{j=1}^{m}g\left(T{E}_j,\frac{\tau }{m}{E}_j\right),\hfill \end{array}$$

that is,

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

(3.4)

Now, using equation (2.2), we have

$$\rho \left(TX-\frac{\tau }{m}X\right)=\left({\nabla }_X\omega -\frac{\tau }{m}\rho X\right),$$

which in view of a local frame $\left\{F_1,..,F_m\right\}$ on $(N^m,g)$ implies

$$\begin{array}{ccc}\hfill {\rho }^2{∥T-\frac{\tau }{m}I∥}^2& =& \sum _{j=1}^{m}g\left(\rho \left(T{E}_j-\frac{\tau }{m}{E}_j\right),\rho \left(T{E}_j-\frac{\tau }{m}{E}_j\right)\right)\hfill \\ & =& \sum _{j=1}^{m}g\left({\nabla }_{E_j}\omega -\frac{\tau }{m}\rho E_j,{\nabla }_{E_j}\omega -\frac{\tau }{m}\rho E_j\right)\hfill \\ & =& {∥\nabla \omega ∥}^2+\frac{1}{m}{\tau }^2{\rho }^2-\frac{2}{m}\tau \rho div\omega .\hfill \end{array}$$

Using (3.2), in above equation, yields

$${\rho }^2{∥T-\frac{\tau }{m}I∥}^2={∥\nabla \omega ∥}^2-\frac{1}{m}{\tau }^2{\rho }^2,$$

which on integration gives

$${\int }_{N^m}{\rho }^2{∥T-\frac{\tau }{m}I∥}^2={\int }_{N^m}\left({∥\nabla \omega ∥}^2-\frac{1}{m}{\tau }^2{\rho }^2\right).$$

(3.5)

Next, we recall the following integral formula (cf. [16])

$${\int }_{N^m}\left(Ric\left(\omega ,\omega \right)+\frac{1}{2}{\left|{\pounds }_{\omega }g\right|}^2-{∥\nabla \omega ∥}^2-{\left(div\omega \right)}^2\right)=0,$$

and employing it in equation (3.5), we conclude

$${\int }_{N^m}{\rho }^2{∥T-\frac{\tau }{m}I∥}^2={\int }_{N^m}\left(Ric\left(\omega ,\omega \right)+\frac{1}{2}{\left|{\pounds }_{\omega }g\right|}^2-{\left(div\omega \right)}^2-\frac{1}{m}{\tau }^2{\rho }^2\right).$$

Using equations (3.2) and (3.3) in above equation, yields

$${\int }_{N^m}{\rho }^2{∥T-\frac{\tau }{m}I∥}^2={\int }_{N^m}\left(Ric\left(\omega ,\omega \right)+2{\rho }^2{∥T∥}^2-{\tau }^2{\rho }^2-\frac{1}{m}{\tau }^2{\rho }^2\right),$$

that is,

$${\int }_{N^m}{\rho }^2{∥T-\frac{\tau }{m}I∥}^2={\int }_{N^m}\left(Ric\left(\omega ,\omega \right)+2{\rho }^2\left({∥T∥}^2-\frac{1}{m}{\tau }^2{\rho }^2\right)-{\tau }^2{\rho }^2+\frac{1}{m}{\tau }^2{\rho }^2\right).$$

In view of equation (3.4), above equation implies

$${\int }_{N^m}{\rho }^2{∥T-\frac{\tau }{m}I∥}^2={\int }_{N^m}\left(\frac{m-1}{m}{\tau }^2{\rho }^2-Ric\left(\omega ,\omega \right)\right)$$

and substituting from equation (3.2), it yields

$${\int }_{N^m}{\rho }^2{∥T-\frac{\tau }{m}I∥}^2=\frac{m-1}{m}{\int }_{N^m}{\left(div\omega \right)}^2-{\int }_{N^m}Ric\left(\omega ,\omega \right).$$

Employing inequality (3.1) in above equation, we conclude

$${\rho }^2{∥T-\frac{\tau }{m}I∥}^2=0.$$

However, $\rho \ne 0$ on connected $N^m$, gives

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

(3.6)

Taking covariant derivative in above equation, we have

$$\left({\nabla }_{X}T\right)\left(Y\right)=\frac{1}{m}X\left(\tau \right)Y$$

and using a frame $\left\{F_1,..,F_m\right\}$ on $(N^m,g)$ in above equation, we get

$$\sum _{j=1}^m\left({\nabla }_{E_j}T\right)\left(E_j\right)=\frac{1}{m}\nabla \tau .$$

Using equation (2.5) in this equation, we arrive at

$$\frac{1}{2}\nabla \tau =\frac{1}{m}\nabla \tau$$

and as $m>2$, we conclude $\nabla \tau =0$. Hence, the scalar curvature $\tau$ is a constant and it is a nonzero constant. Now, equations (2.7) and (3.6) imply

$$\frac{\tau }{m}\omega =\frac{\tau }{m}\nabla \rho -\tau \nabla \rho ,$$

that is,

$$\omega =-(m-1)\nabla \rho$$

(3.7)

and it gives $div\omega =-(m-1)\Delta \rho$, which in view of equation (3.2) implies $\tau \rho =-(m-1)\Delta \rho$, that is,

$$-(m-1)\rho \Delta \rho =\tau {\rho }^2.$$

Integrating above equation by parts, we arrive at

$$(m-1){\int }_{N^m}{∥\nabla \rho ∥}^2=\tau {\int }_{N^m}{\rho }^2.$$

Since, $\rho$ is a nonconstant, from above equation, we conclude the constant $\tau >0$. We put $\tau =m(m-1)\alpha$ for a positive constant $\alpha$. Now, differentiating equation (3.7) and using equations (2.2) and (3.6), we conclude

$${\nabla }_X\nabla \rho =-\alpha \rho X,X\in \Theta \left(T{N}^m\right),$$

where $\rho$ is nonconstant function and $\alpha >0$ is a constant. Hence, $Hess\left(\rho \right)=-\alpha \rho g$, that is, $(N^m,g)$ is isometric to the sphere $S^m\left(\alpha \right)$ (cf. [14,15]).

Conversely, suppose that $(N^m,g)$ is isometric to the sphere $S^m\left(\alpha \right)$. Then, we know that a nonzero constant vector field $\mathbf{b}$ on the ambient Euclidean space $R^{m+1}$ induces a vector field $\omega$ on the sphere $S^m\left(\alpha \right)$, which by equation (1.4) is a $\rho$-Ricci vector field. Clearly, the scalar curvature of $S^m\left(\alpha \right)$ is given by $\tau =m(m-1)\alpha \ne 0$. We claim that the function $\rho$ is nonzero and nonconstant. If $\rho =0$, then by equation (1.4), we have $f=0$, which in view of equation (1.3) implies $\omega =0$, and this in turn will imply that the constant vector field $\mathbf{b}=0$. This is a contrary to the assumption that $\mathbf{b}$ is a nonzero constant vector field. Hence, $\rho \ne 0$. Now, suppose $\rho$ is a constant, then by equation (1.4), f is a constant and by equation (1.3), we have $div\omega =-m\sqrt{\alpha }f$, which by Stokes’s Theorem on compact $S^m\left(\alpha \right)$, would imply $f=0$. This in turn by virtue of equation (1.4) implies $\rho =0$, which is a contradiction as seen above. Hence, the function $\rho$ is nonzero and nonconstant.

Next, using equations (1.3) and (1.4), we have

$$div\omega =m(m-1)\alpha \rho$$

(3.8)

and it gives

$${\int }_{S^m\left(\alpha \right)}{\left(div\omega \right)}^2=m^2{(m-1)}^2{\alpha }^2{\int }_{S^m\left(\alpha \right)}{\rho }^2.$$

(3.9)

Now, using equation (1.4), we have

$$\nabla \rho =-\frac{1}{\left(m-1\right)\sqrt{\alpha }}\nabla f,$$

(3.10)

which on using equation (1.3), gives

$$\nabla \rho =-\frac{1}{m-1}\omega .$$

Taking divergence in above equation and using equation (3.8), we conclude $\Delta \rho =-m\alpha \rho$, that is, $\rho \Delta \rho =-m\alpha {\rho }^2$ integrating this equation by parts, we conclude

$${\int }_{S^m\left(\alpha \right)}{∥\nabla \rho ∥}^2=m\alpha {\int }_{S^m\left(\alpha \right)}{\rho }^2.$$

Treating this equation with equation (3.9), we conclude

$${\int }_{S^m\left(\alpha \right)}{\left(div\omega \right)}^2=m{(m-1)}^2\alpha {\int }_{S^m\left(\alpha \right)}{∥\nabla \rho ∥}^2.$$

(3.11)

Also, using equations (1.3) and (3.10), we have

$$\omega =-(m-1)\nabla \rho$$

and it changes the equation (3.11) to

$${\int }_{S^m\left(\alpha \right)}{\left(div\omega \right)}^2=m\alpha {\int }_{S^m\left(\alpha \right)}{∥\omega ∥}^2.$$

Finally, using $Ric\left(\omega ,\omega \right)=(m-1){∥\omega ∥}^2$ in above equation, we conclude

$${\int }_{S^m\left(\alpha \right)}Ric\left(\omega ,\omega \right)=\frac{m-1}{m}{\int }_{S^m\left(\alpha \right)}{\left(div\omega \right)}^2$$

and this finishes the proof. □

Next, we consider a closed $\rho$-Ricci vector field on a compact and connected Riemannian manifold $\left(N^m,g\right)$ such that the smooth function $\rho$ is a nontrivial solution of the FM-equation and find yet another characterization of the sphere $S^m\left(\alpha \right)$. Indeed we prove the following:

Theorem 2.

An m-dimensional complete and simply connected Riemannian manifold $\left(N^m,g\right)$ with scalar curvature $\tau >0$ admits a closed ρ-Ricci vector field ω such that the function ρ is a nontrivial solution of the FM-equation and the length of covariant derivative of ω satisfies

$${∥\nabla \omega ∥}^2\le \frac{1}{m}{\tau }^2{\rho }^2,$$

if and only if, τ is a positive constant $\tau =m(m-1)\alpha$ and $\left(N^m,g\right)$ is isometric to $S^m\left(\alpha \right)$.

Proof. 

Suppose $\left(N^m,g\right)$ is an m-dimensional complete and simply connected Riemannian manifold with scalar curvature $\tau >0$, and it admits a closed $\rho$-Ricci vector field $\omega$, where $\rho$ is nontrivial solution of the FM-equation (1.6) and the length of covariant derivative of $\omega$ satisfies

$${∥\nabla \omega ∥}^2\le \frac{1}{m}{\tau }^2{\rho }^2.$$

(3.12)

For $\rho$, we define the operator $B_{\rho }$ by

$$B_{\rho }X={\nabla }_X\nabla \rho ,X\in \Theta \left(T{N}^m\right),$$

then $B_{\rho }$ is a symmetric operator related to $Hess\left(\rho \right)$ by

$$Hess\left(\rho \right)\left(X,Y\right)=g\left(B_{\rho }X,Y\right),X,Y\in \Theta \left(T{N}^m\right).$$

(3.13)

As, $\rho$ is a nontrivial solution of the FM-equation, using equations (3.13) and (1.6), we have

$$\rho TX=B_{\rho }X-\left(\Delta \rho \right)X,$$

which in view of equation (1.7) becomes

$$B_{\rho }X=\rho TX-\frac{\tau }{m-1}\rho X.$$

(3.14)

Note that owing to the fact that $\rho$ is a nontrivial solution of the FM-equation on $\left(N^m,g\right)$, the scalar curvature $\tau$ is a constant and we put $\tau =m(m-1)\alpha$ for a constant $\alpha$. Using equation (3.14), we have

$$B_{\rho }X+\alpha \rho X=\rho TX-(m-1)\alpha \rho X,X\in \Theta \left(T{N}^m\right).$$

Now, using equation (2.2) in above equation, we get

$$B_{\rho }X+\alpha \rho X={\nabla }_X\omega -(m-1)\alpha \rho X,X\in \Theta \left(T{N}^m\right).$$

Taking a local frame $\left\{F_1,..,F_m\right\}$ on $\left(N^m,g\right)$, by above equation, we conclude

$$\begin{array}{ccc}\hfill {∥B_{\rho }+\alpha \rho I∥}^2& =& \sum _{j=1}^{m}g\left(B_{\rho }{F}_j+\alpha \rho F_j,B_{\rho }{F}_j+\alpha \rho F_j\right)\hfill \\ & =& \sum _{j=1}^{m}g\left({\nabla }_{F_j}\omega -(m-1)\alpha \rho F_j,{\nabla }_{F_j}\omega -(m-1)\alpha \rho F_j\right)\hfill \\ & =& {∥\nabla \omega ∥}^2+m{(m-1)}^2{\alpha }^2{\rho }^2-2(m-1)\alpha \rho \left(div\omega \right).\hfill \end{array}$$

Now, using equating (2.2), we have $div\omega =\tau \rho =m(m-1)\alpha \rho$ and inserting it in above equation, we arrive at

$${∥B_{\rho }+\alpha \rho I∥}^2={∥\nabla \omega ∥}^2-m{(m-1)}^2{\alpha }^2{\rho }^2,$$

that is,

$${∥B_{\rho }+\alpha \rho I∥}^2={∥\nabla \omega ∥}^2-\frac{1}{m}{\tau }^2{\rho }^2.$$

Using inequality (3.12) in above equation results in

$$B_{\rho }=-\alpha \rho I,$$

that is,

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

(3.15)

Note that as $\tau >0$, the constant $\alpha >0$ and $\rho$ being a nontrivial solution, $\rho$ is a nonconstant function. Hence, by equation (3.15), the complete and simply connected Riemannian manifold $\left(N^m,g\right)$ is isometric to the sphere $S^m\left(\alpha \right)$ (cf. [14,15]).

Conversely, suppose that $\left(N^m,g\right)$ is isometric to the sphere $S^m\left(\alpha \right)$. Then, by equation (1.7), the function f is a solution of FM-equation on the sphere $S^m\left(\alpha \right)$, which has a closed $\rho$-Ricci vector field $\omega$. The solution f of FM-equation is related to $\rho$ by equation (1.4), that is,

$$f=-(m-1)\sqrt{\alpha }\rho .$$

(3.16)

In the proof of Theorem-1, we have seen that $\rho$ is a nonconstant function on $S^m\left(\alpha \right)$. Moreover, using equation (3.16), we have

$$\Delta f=-(m-1)\sqrt{\alpha }\Delta \rho ,Hess\left(f\right)=-(m-1)\sqrt{\alpha }Hess\left(\rho \right)$$

and the equation (1.7) takes the form

$$-(m-1)\sqrt{\alpha }\left(\Delta \rho \right)g+fRic=-(m-1)\sqrt{\alpha }Hess\left(\rho \right),$$

which in view of equation (3.16) changes to

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

Hence, $\rho$ is a nontrivial solution of the FM-equation on the sphere $S^m\left(\alpha \right)$. Now, the Ricci operator T of the sphere $S^m\left(\alpha \right)$ is given by $T=(m-1)\alpha I$ and therefore equation (2.2) on $S^m\left(\alpha \right)$ is

$${\nabla }_X\omega =(m-1)\alpha \rho X,X\in \Theta \left(T{S}^m\left(\alpha \right)\right).$$

Using the expression for the scalar curvature $\tau =m(m-1)\alpha$ for the sphere $S^m\left(\alpha \right)$, we have

$${\nabla }_X\omega =\frac{\tau }{m}\rho X,X\in \Theta \left(T{S}^m\left(\alpha \right)\right).$$

This proves

$${∥\nabla \omega ∥}^2=\frac{1}{m}{\tau }^2{\rho }^2$$

and completes the proof. □

4. Conclusions

In previous section, we have used a closed $\rho$-Ricci vector field $\omega$ on an m-dimensional Riemannian manifold $\left(N^m,g\right)$ to find two different characterizations of a m-sphere $S^m\left(\alpha \right)$. The scope of studying $\rho$-Ricci vector fields on a Riemannian manifold is quite modest. We observe that, in previous section, we restricted the $\rho$-Ricci vector field $\omega$ to be closed that simplified the expression for the covariant derivative of $\omega$. It will be interesting to investigate whether, we could get similar results after removing the restriction that the $\rho$-Ricci vector field $\omega$ is closed. It will be interesting future topic to study the geometry of an m-dimensional Riemannian manifold $\left(N^m,g\right)$ that admits a $\rho$-Ricci vector field $\omega$, which need not be closed. In order to simplify the findings on an m-dimensional Riemannian manifold $\left(N^m,g\right)$ admitting a $\rho$-Ricci vector field $\omega$, which is not necessarily closed, we could impose the restriction on the Ricci operator T of $\left(N^m,g\right)$ to be Codazzi type tensor, namely it satisfies

$$\left({\nabla }_{X}T\right)\left(Y\right)=\left({\nabla }_{Y}T\right)\left(X\right),X,Y\in \Theta \left(T{N}^m\right).$$

Note that above restriction on $\left(N^m,g\right)$ is slightly stronger than demanding the scalar curvature is a constant.