Characterizations of Spheres and Euclidean Spaces by Conformal Vector Fields

Abstract

A nontrivial conformal vector field $\omega$ on an m-dimensional connected Riemannian manifold $\left(M^m,g\right)$ has naturally associated with it the conformal potential $\theta$, a smooth function on $M^m$, and a skew-symmetric tensor T of type $(1,1)$ called the associated tensor. There is a third entity, namely the vector field $T\omega$, called the orthogonal reflection field, and in this article, we study the impact of the condition that commutator $\left[\omega ,T\omega \right]=0$; this condition that we refer to as the orthogonal reflection field is commutative. As a natural impact of this condition, we see the existence of a smooth function $\rho$ on $M^m$ such that $\nabla \theta =\rho \omega$; this function $\rho$ is called the proportionality function. First, we show that an m-dimensional compact and connected Riemannian manifold $\left(M^m,g\right)$ admits a nontrivial conformal vector field $\omega$ with a commuting orthogonal reflection $T\omega$ and constant proportionality function $\rho$ if and only if $\left(M^m,g\right)$ is isometric to the sphere $S^m\left(c\right)$ of constant curvature c. Secondly, we find three more characterizations of the sphere and two characterizations of a Euclidean space using these ideas. Finally, we provide a condition for a conformal vector field on a compact Riemannian manifold to be closed.

1. Introduction

Conformal geometry is an interesting branch of differential geometry for all times, and one can trace its interest through old articles, such as [1]. It evolved over time and was constantly becoming enriched as we find in the most recent work [2]. In conformal geometry, an important topic is studying the influence of a conformal vector field $\omega$ on an m-dimensional Riemannian manifold $(M^m,g)$. We shall abbreviate a conformal vector field $\omega$ as CONFLVF $\omega$ for the sake of convenience. There is a smooth function $\theta$ that is naturally associated with a CONFLVF $\omega$ on $(M^m,g)$ called the conformal potential satisfying

$${\displaystyle \frac{1}{2}}{\pounds }_{\omega }g=\theta g,$$

(1)

where £ is the Lie derivative operator. A CONFLVF $\omega$ is said to be Killing if conformal potential $\theta =0$ and consequently, a nontrivial CONFLVF $\omega$ must have conformal potential $\theta \ne 0$. There is a $(1,1)$ skew-symmetric tensor field T naturally associated with a CONFLVF $\omega$ on $(M^m,g)$ called the associated tensor of CONFLVF $\omega$, defined by

$${\displaystyle \frac{1}{2}}d\alpha (X,Y)=g\left(TX,Y\right),$$

(2)

for smooth vector fields $X,Y$ on $M^m$, where $\alpha$ is 1-form dual to $\omega$. This associated tensor T plays a crucial role in studying the impact of a CONFLVF $\omega$ on the geometry of $(M^m,g)$ (cf. [2]).

The sphere $S^m\left(c\right)$ of constant curvature c as a hypersurface of the Euclidean space $\left(E^{m+1},〈,〉\right)$, where $〈,〉$ is the Euclidean metric, has a unit normal $\zeta$, induced metric g and the shape operator $A=-\sqrt{c}I$. On choosing a unit constant vector field Z on the Euclidean space $\left(E^{m+1},〈,〉\right)$, its tangential component $\omega$ to the sphere $S^m\left(c\right)$ satisfies

$${\nabla }_X\omega =-\sqrt{c}fX\mathrm{and}\nabla f=\sqrt{c}\omega ,$$

(3)

where $f=〈Z,\zeta 〉$ and $\nabla f$ is the gradient of f on $S^m\left(c\right)$ with respect to the induced metric g. Thus, we see that $\omega$ is a CONFLVF on the sphere $S^m\left(c\right)$ with conformal potential $\theta =-\sqrt{c}f$. This CONFLVF $\omega$ is closed and therefore the associated tensor $T=0$.

Consider the unit sphere $S^{2m-1}$ as the hypersurface of the Euclidean space $\left(E^{2m},〈,〉\right)$ with unit normal $\zeta$ and CONFLVF $\omega$ induced by a constant unit vector Z on $\left(E^{2m},〈,〉\right)$, as described in the previous paragraph with a conformal potential $\theta =-f$. Also, using the complex structure J on $\left(E^{2m},〈,〉\right)$, define a unit vector field $\xi =J\zeta$, which has covariant derivative

$${\nabla }_X\xi ={\left(JX\right)}^T,$$

(4)

where ${\left(JX\right)}^T$ is the tangential component of $JX$ to $S^{2m-1}$ and we see that $\xi$ is a Killing vector field, that is,

$${\pounds }_{\xi }g=0.$$

Now, defining a vector field $\overline{\omega }=\omega +\xi$, we obtain a CONFLVF $\overline{\omega }$ with conformal potential $\theta =-f$, which is not closed and indeed has associated operator $TX={\left(JX\right)}^T$. Non-closed CONFLVFs are in abundance, for instance on the Euclidean space $\left(E^{2m},〈,〉\right)$; if $\xi$ is the position vector field on $E^{2m}$, then

$$\omega =\xi +J\xi$$

is a CONFLVF on $\left(E^{2m},〈,〉\right)$, which is not closed and has associated tensor $TX=JX$.

Also, consider the compact Riemannian manifold $\left(M^m,g\right)$, where $M^m=S^1{\times }_h{S}^{m-1}\left(c\right)$ is the warped product, where h is a smooth positive function on the unit circle $S^1$ and the warped product metric $g=d{t}^2+h^2\overline{g}$, t is coordinate function on $S^1$ and $\overline{g}$ is the canonical metric on the sphere $S^{m-1}\left(c\right)$ of constant curvature c. Then, the vector field $\omega =h{\displaystyle \frac{\partial }{\partial t}}$ on $\left(M^m,g\right)$ satisfies (cf. [3])

$${\nabla }_X\omega =h^{\prime }X,$$

where X is any vector field on $M^m$. Thus, we obtain

$${\displaystyle \frac{1}{2}}{\pounds }_{X}g=\theta g,$$

that is, $\omega$ is a closed CONFLVF $\left(M^m,g\right)$ with conformal potential $\theta =h^{\prime }$ and associated tensor $T=0$. We have following expression for the Ricci operator S on the warped product manifold $(M^m,g)$ (cf. [4])

$$S\left(X\right)=-{\displaystyle \frac{n-1}{h}}{\mathcal{H}}^{h}X,$$

(5)

for horizontal vector field X on $S^1$, where ${\mathcal{H}}^h$ is the Hessian operator of h and

$$S\left(V\right)=(n-2)cV-\left({\displaystyle \frac{h^{″}}{h}}+(n-2){\left({\displaystyle \frac{h^{″}}{h}}\right)}^2\right)V$$

for vertical vector field V on $S^{m-1}\left(c\right)$. As the CONFLVF $\omega$ is horizontal, we see by Equation (5) that

$$S\left(\omega \right)=-(n-1){\mathcal{H}}^h\left({\displaystyle \frac{\partial }{\partial t}}\right)=-(n-1)\nabla \sigma ,$$

where $\nabla \sigma$ is the gradient of $\sigma$. The scalar curvature $\tau$ of $\left(M^m,g\right)$ is given by

$$\tau =-{\displaystyle \frac{n-1}{h^2}}\left(2h{h}^{″}+(n-2)h^{\prime }-(n-2)c\right).$$

Riemannian manifolds admitting closed CONFLVF have been studied quite extensively (cf. [5,6,7,8,9,10,11,12,13]). Riemannian manifolds with non-closed CONFLVF have been studied in [14,15,16,17]. Moreover, apart from the fact that the presence of a CONFLVF on a Riemannian manifold influences its geometry, they are also used in theory of relativity (cf. [18,19,20,21]). Note that the study of submanifolds of Euclidean spaces becomes convenient due presence of position vector field, and taking the clue that the position vector field is a closed CONFLVF, there was a fashion for studying submanifolds of Riemannian manifolds (non-Euclidean spaces) which possess closed CONFLVF; this gave another role to CONFLVF (cf. [5] and references therein).

Suppose that $\omega$ is a CONFLVF on a Riemannian manifold $(M^m,g)$ with associated tensor T. As the vector $T\omega$ is orthogonal to $\omega$, we call the vector $T\omega$ the orthogonal reflection of $\omega$ and further, if it satisfies $\left[\omega ,T\omega \right]=0$, we call the vector $T\omega$ a commutative orthogonal reflection. It is clear that closed CONFLVF $\omega$ has commutative orthogonal reflections and non-closed CONFLVF $\omega$ satisfying $T\omega =0$ has a commutative orthogonal reflection. In this article, first we notice that for a CONFLVF $\omega$ with a commutative orthogonal reflection, there exists a smooth function $\rho$ satisfying $\nabla \theta =\rho \omega$, where $\theta$ is the conformal potential of $\omega$. This function $\rho$ is called the proportionality function for the CONFLVF $\omega$ with commuting orthogonal reflection. Observe that on the sphere $S^m\left(c\right)$ with a nontrivial CONFLVF $\omega$ satisfying Equation (9), which is closed and therefore has commutative orthogonal reflection and through Equation (3), we have $\nabla \theta =-\omega$, that is, the proportionality function of this CONFLVF on $S^m\left(c\right)$ is $\rho =-c$. This raises a question: is an m-dimensional compact and connected Riemannian manifold $(M^m,g)$ that admits a nontrivial CONFLVF $\omega$ with a commuting orthogonal reflection having proportionality function $\rho$ as a constant necessarily isometric to $S^m\left(c\right)$? In this paper, we show that the answer to this question is in affirmative and obtain a new characterization of the sphere $S^m\left(c\right)$ (see Proposition 1).

Also, observe that the scalar curvature $\tau$ of the sphere $S^m\left(c\right)$ is given by $\tau =n(n-1)c$ and we see that the conformal potential $\theta$ of the CONFLVF $\omega$ on the sphere $S^m\left(c\right)$ described in Equation (3) satisfies

$$\nabla \theta =-{\displaystyle \frac{\tau }{n(n-1)}}\xi .$$

(6)

This motivates the following question: under what conditions is a compact and connected Riemannian manifold $(M^m,g)$ admitting a nontrivial CONFLVF $\omega$ with conformal potential $\theta$ satisfying Equation (6) isometric to the sphere $S^m\left(c\right)$? We answer this question and find yet another characterization of the sphere $S^m\left(c\right)$ (see Theorem 1).

Next, we consider a nontrivial CONFLVF $\omega$ with conformal potential $\theta$, commutative orthogonal reflection on an m-dimensional compact and connected Riemannian manifold $(M^m,g)$ such that the proportionality function $\rho$ is a constant along the integral curves of $\omega$ and show that under the condition that the integral of the Ricci curvature $\mathrm{Ric}\left(\nabla \theta ,\nabla \theta \right)$ has a suitable lower bound, $\left(M^m,g\right)$ necessarily isometric to $S^m\left(c\right)$. The converse also holds (see Theorem 2). Finally, we show that an m-dimensional compact and connected Riemannian manifold $\left(M^m,g\right)$ with Ricci operator S and scalar curvature $\tau$ admitting a nontrivial CONFLVF $\omega$ with orthogonal reflection field $T\omega =0$, $S\left(\omega \right)=\lambda \omega$ for a nonzero constant $\lambda$ and scalar curvature $\tau$ constant along the integral curves of $\omega$ is necessarily isometric to $S^m\left(c\right)$ and the converse also holds (see Theorem 3).

Next, the paper ends with two characterizations of the Euclidean space $\left(E^n,〈,〉\right)$. In the first result we show that for an m-dimensional complete and connected Riemannian manifold $\left(M^m,g\right)$ to be isometric to the Euclidean space $\left(E^n,〈,〉\right)$, it is necessary and sufficient that it admits a nontrivial CONFLVF $\omega$ with conformal potential $\theta$, orthogonal reflection $T\omega =0$ and the proportionality function $\rho$ satisfies $\rho \mathrm{Ric}\left(\omega ,\omega \right)\ge 0$ (see Theorem 4). In the last result we prove that m-dimensional complete and connected Riemannian manifold $\left(M^m,g\right)$ admits a nontrivial CONFLVF $\omega$ with conformal potential $\theta$ constant along integral curves of $\omega$ and orthogonal reflection $T\omega =0$ if and only if $\left(M^m,g\right)$ is isometric to the Euclidean space $\left(E^n,〈,〉\right)$.

2. Preliminaries

Let $\omega$ be a CONFLVF on an m-dimensional Riemannian manifold $(M^m,g)$ with conformal potential $\theta$ obeying

$${\displaystyle \frac{1}{2}}{\pounds }_{\omega }g=\theta g.$$

(7)

Note that if $\theta =0$, then CONFLVF $\omega$ is Killing. Therefore, we say $\omega$ is a nontrivial CONFLVF if the conformal potential $\theta \ne 0$. We denote by $\alpha$ the 1-form dual to CONFLVF $\omega$, that is, $\alpha \left(X\right)=g\left(X,\omega \right)$ for any smooth vector field X on $M^m$. Then, there is a naturally associated skew-symmetric $\left(1,1\right)$ tensor T to $\omega$ defined by

$${\displaystyle \frac{1}{2}}d\alpha \left(X,Y\right)=g\left(TX,Y\right),$$

(8)

where $X,Y$ are arbitrary smooth vector fields on $M^m$. This skew symmetric tensor T associated with the CONFLVF $\omega$ is called the associated tensor and it plays an important role in our study. We denote the Levi–Civita connection on $\left(M^m,g\right)$ by ∇ and observe that

$$\left({\pounds }_{\omega }g\right)\left(X,Y\right)=g\left({\nabla }_X\omega ,Y\right)+g\left({\nabla }_Y\omega ,X\right)$$

and

$$d\alpha \left(X,Y\right)=g\left({\nabla }_X\omega ,Y\right)-g\left({\nabla }_Y\omega ,X\right).$$

Consequently, we conclude

$$2g\left({\nabla }_X\omega ,Y\right)=\left({\pounds }_{\omega }g\right)\left(X,Y\right)+d\alpha \left(X,Y\right),$$

which, on utilizing Equations (7) and (8), yields

$${\nabla }_X\omega =\theta X+TX$$

(9)

for any vector field X on $M^m$. We use the covariant derivative

$$\left({\nabla }_{X}T\right)\left(Y\right)={\nabla }_{X}TY-T\left({\nabla }_{X}Y\right),$$

with Equation (9), in order to compute the following expression for the curvature tensor R of $\left(M^m,g\right)$

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

(10)

where $X,Y$ are arbitrary smooth vector fields $M^m$. In order to compute the Ricci tensor $\mathrm{Ric}$ from above equation, we chose a orthonormal local frame $\left\{v_1,\cdots ,v_m\right\}$ on $(M^m,g)$. Then, we obtain

$$\mathrm{Ric}(X,\omega )=-(n-1)X\left(\theta \right)-g\left(X,\mathrm{div}T\right),$$

(11)

where

$$\mathrm{div}T=\sum _k\left({\nabla }_{v_k}T\right)\left(v_k\right).$$

(12)

Thus, for CONFLVF $\omega$ on an m-dimensional Riemannian manifold $(M^m,g)$ with conformal potential $\theta$, on using Equation (11), we find the following expression for the Ricci operator S

$$S\left(\omega \right)=-(m-1)\nabla \theta -\mathrm{div}T,$$

(13)

where $\mathrm{Ric}\left(X,Y\right)=g\left(SX,Y\right)$ and $\nabla \theta$ is the gradient of the conformal potential $\theta$. On $(M^m,g)$, the scalar curvature $\tau =Tr.S$ satisfies the following (cf. [3,12])

$${\displaystyle \frac{1}{2}}\nabla \tau =\mathrm{div}S,$$

(14)

where

$$\mathrm{div}S=\sum _k\left({\nabla }_{v_k}S\right)\left(v_k\right).$$

Using Equation (7), we obtain the following expression for the divergence of the CONFLVF $\omega$ defined on $(M^m,g)$ with conformal potential $\theta$,

$$\mathrm{div}\omega =m\theta .$$

(15)

On looking at the definition of the associated tensor T of a CONFLVF $\omega$ on $(M^m,g)$ with conformal potential $\theta$, we see that the differential 2-form $g\left(TX,Y\right)$ is closed and therefore we have

$$g\left(\left({\nabla }_{X}T\right)\left(Y\right),Z\right)+g\left(\left({\nabla }_{Y}T\right)\left(Z\right),X\right)+g\left(\left({\nabla }_{Z}T\right)\left(X\right),Y\right)=0,$$

for vector fields $X,Y,Z$ on $M^m$, and on using skew symmetry of the associated tensor T together with Equation (10) in the above equation, we conclude

$$g\left(R(X,Y)\omega -X\left(\theta \right)Y+Y\left(\theta \right)X,Z\right)+g\left(\left({\nabla }_{Z}T\right)\left(X\right),Y\right)=0$$

and it immediately yields

$$\left({\nabla }_{X}T\right)\left(Y\right)=R\left(X,\omega \right)Y+Y\left(\theta \right)X-g\left(X,Y\right)\nabla \theta .$$

(16)

Definition 1.

Given a CONFLVF ω on $(M^m,g)$ with conformal potential θ and associated tensor T, we call the vector field $T\omega$, being orthogonal to ω, the orthogonal reflection field. Also, we say the orthogonal reflection field $T\omega$ is commutative if the Lie bracket $\left[\omega ,T\omega \right]=0$.

In this article, we will focus on the impact of the presence of a CONFLVF $\omega$ with commutative orthogonal reflection field $T\omega$ on an m-dimensional Riemannian manifold $\left(M^m,g\right)$.

Lemma 1.

The orthogonal reflection field $T\omega$ of a nontrivial CONFLVF ω with conformal potential θ on an m-dimensional Riemannian manifold $\left(M^m,g\right)$ is commutative if and only if there is a smooth function ρ on $M^m$ satisfying

$$\nabla \theta =\rho \omega .$$

Proof. 

Suppose that the orthogonal reflection field $T\omega$ is commutative, that is,

$$\left[\omega ,T\omega \right]=0.$$

(17)

Now, using Equation (9), we compute

$${\nabla }_{\omega }T\omega =\left({\nabla }_{\omega }T\right)\left(\omega \right)+T\left(\theta \omega +T\omega \right)$$

and inserting an appropriate form of Equation (16) in the above equation yields

$${\nabla }_{\omega }T\omega =\omega \left(\theta \right)\omega -{∥\omega ∥}^2\nabla \theta +\theta T\omega +T^2\omega .$$

(18)

Also, on using Equation (9), we have

$${\nabla }_{T\omega }\omega =\theta T\omega +T\left(T\omega \right)=\theta T\omega +T^2\omega .$$

(19)

Combining Equations (17)–(19), we obtain

$${∥\omega ∥}^2\nabla \theta =\omega \left(\theta \right)\omega$$

and taking the inner product by $\nabla \theta$ in above equation leads to

$${∥\omega ∥}^2{∥\nabla \theta ∥}^2=\omega {\left(\theta \right)}^2=g{\left(\omega ,\nabla \theta \right)}^2.$$

(20)

Note that above equation is equality in the Cauchy–Schwarz inequality

$$g{\left(\omega ,\nabla \theta \right)}^2\le {∥\omega ∥}^2{∥\nabla \theta ∥}^2$$

and therefore equality (20) holds if and only if vector fields $\omega ,\nabla \theta$ are parallel. Hence, there exists a smooth function $\rho$ on $M^m$ such that

$$\nabla \theta =\rho \omega .$$

(21)

Conversely, if (21) holds, then Equations (18), (19) and (21) imply (17) and that the orthogonal reflection $T\omega$ is commutative. □

Observe that for a CONFLVF $\omega$ on a Riemannian manifold $\left(M^m,g\right)$ with conformal potential $\theta$ and associated tensor T; if $T\omega =0$, then automatically the orthogonal reflection field $T\omega$ is commutative and in this case as a particular case of Lemma 1, giving us the following one-way result.

Corollary 1.

If the orthogonal reflection field $T\omega =0$ for a nontrivial CONFLVF ω with conformal potential θ on an m-dimensional Riemannian manifold $\left(M^m,g\right)$, then there is a smooth function ρ on $M^m$ satisfying

$$\nabla \theta =\rho \omega .$$

Definition 2.

If the CONFLVF ω on a Riemannian manifold $\left(M^m,g\right)$ with conformal potential θ and orthogonal reflection field $T\omega$ is commutative, then the function ρ appearing in Lemma 1 is called proportionality function.

Also, we compute the divergence of the vector field $S\omega$ as follows,

$$\mathrm{div}S\omega =\sum _{k}g\left({\nabla }_{v_k}S\omega ,v_k\right)=\sum _k\left(\left({\nabla }_{v_k}S\right)\left(\omega \right)+S\left({\nabla }_{v_k}\omega \right),v_k\right),$$

which on using symmetry of S and Equations (9) and (14), yields

$$\mathrm{div}S\omega ={\displaystyle \frac{1}{2}}\omega \left(\tau \right)+\sum _{k}g\left(S\left(v_k\right),\theta v_k+T{v}_k\right)$$

and T being skew-symmetric, we reach

$$\mathrm{div}S\omega ={\displaystyle \frac{1}{2}}\omega \left(\tau \right)+\theta \tau .$$

(22)

Similarly, on using Equation (9) and skew symmetry of the associated operator T, for the CONFLVF $\omega$ on an n-dimensional Riemannian manifold $(M^m,g)$ with conformal potential $\theta$, we find

$$\mathrm{div}\left(T\omega \right)=-{∥T∥}^2-g\left(\omega ,\mathrm{div}T\right),$$

(23)

where

$${∥T∥}^2=\sum _{k}g\left(T{v}_k,T{v}_k\right).$$

Now, for the conformal potential $\theta$ of the CONFLVF $\omega$ on an n-dimensional Riemannian manifold $(M^m,g)$, the Hessian $\mathrm{Hess}\left(\theta \right)$ of $\theta$ is the symmetric bilinear form defined by

$$\mathrm{Hess}\left(\theta \right)\left(X,Y\right)=X\left(Y\theta \right)-\left({\nabla }_{X}Y\right)\left(\theta \right)$$

and the Hessian operator ${\mathcal{H}}^{\theta }$ of $\theta$ is defined by

$$\mathrm{Hess}\left(\theta \right)\left(X,Y\right)=g\left({\mathcal{H}}^{\theta }X,Y\right)$$

for smooth vector fields $X,Y$ on $M^m$. The Laplacian $\Delta \theta$ of $\theta$ is defined by $\Delta \theta =\mathrm{div}\left(\nabla \theta \right)$, which is also given by $\Delta \theta =Tr.{\mathcal{H}}^{\theta }$. If $(M^m,g)$ is a compact Riemannian manifold, then we have the following Bochner’s formula

$${\int }_{M^m}\mathrm{Ric}\left(\nabla \theta ,\nabla \theta \right)d{V}_g={\int }_{M^m}\left({\left(\Delta \theta \right)}^2-{∥{\mathcal{H}}^{\theta }∥}^2\right)d{V}_g,$$

(24)

where for a local orthonormal frame $\left\{v_1,\dots ,v_n\right\}$ on $\left(M^m,g\right)$

$${∥{\mathcal{H}}^{\theta }∥}^2=\sum _{k}g\left({\mathcal{H}}^{\theta }{v}_k,{\mathcal{H}}^{\theta }{v}_k\right).$$

The following proposition shows the importance of a commutative orthogonal reflection field $T\omega$ and the proportionality function $\rho$ corresponding to a CONFLVF $\omega$ on a Riemannian manifold $\left(M^m,g\right)$.

Proposition 1.

An m-dimensional compact and connected Riemannian manifold $\left(M^m,g\right)$ admits a nontrivial CONFLVF ω with conformal potential θ, commutative orthogonal reflection field $T\omega$ and proportionality function ρ a constant if and only if constant $\rho =-c$ with $c>0$ and $\left(M^m,g\right)$ isometric to the round sphere $S^m\left(c\right)$.

Proof. 

Suppose the proportionality function $\rho$ is a constant. Then, by Lemma 1, we have

$$\nabla \theta =\rho \omega .$$

(25)

Differentiating above equation with respect a smooth vector field X on $M^m$ while using Equation (9), we arrive at

$${\mathcal{H}}^{\theta }X=\rho \theta X+\rho TX,$$

that is,

$$\left({\mathcal{H}}^{\theta }-\rho \theta I\right)=\rho T.$$

The left hand side in above equation is symmetric, while the right hand side is skew-symmetric and therefore we conclude

$${\mathcal{H}}^{\theta }=\rho \theta I\mathrm{and}\rho T=0$$

(26)

Note that if the constant $\rho =0$, then Equation (25) will imply the conformal potential $\theta$ is a constant, which through the integration of Equation (15) will imply $\theta =0$. This is a contradiction to the fact that $\omega$ is a nontrivial CONFLVF. Thus, the constant $\rho \ne 0$ and by second equation in (26) confirms $T=0$. Taking trace in the first equation in (26) provides $\Delta \theta =n\rho \theta$, that is, $\theta \Delta \theta =n\rho {\theta }^2$, and integrating the last relation by parts gives

$${\int }_{M^m}{∥\nabla \theta ∥}^{2}d{V}_g=-n\rho {\int }_{M^m}{\theta }^{2}d{V}_g$$

and as the conformal potential $\theta$ is non-constant (guaranteed by $\omega$ being nontrivial), the above equation confirms constant $\rho <0$. We take $\rho =-c$, which gives the constant $c>0$. This makes the first equation in (26)

$${\mathcal{H}}^{\theta }=-c\theta I$$

with $\theta$ a non-constant function and positive constant c, Obata’s differential equation (cf. [10,11]). Hence, $\left(M^m,g\right)$ is isometric to $S^m\left(c\right)$.

Conversely, suppose $\left(M^m,g\right)$ is isometric to $S^m\left(c\right)$. Then, through Equation (3), we see that there is a nontrivial CONFLVF $\omega$ on $S^m\left(c\right)$ with conformal potential $\theta$ and as $\omega$ is closed and, therefore, the associated tensor $T=0$ and automatically the orthogonal reflection field are commutative. Now, using Equation (3), we have

$$\nabla \theta =-c\omega ,$$

that is, the proportionality function $\rho =-c$ a constant. Hence, the converse holds. □

3. Characterizations of Spheres

In this section, we consider a nontrivial CONFLVF $\omega$ on an m-dimensional Riemannian manifold $(M^m,g)$ with conformal potential $\theta$ and associated tensor T and focus our attention on the orthogonal reflection field $T\omega$ being commutative, which gives an additional tool namely the proportionality function $\rho$ of Lemma 1. Also, notice if the orthogonal reflection field $T\omega =0$, then it is automatically commutative and in this case we also have the proportionality function $\rho$ given by the Corollary 1, though in this case the converse does not hold unlike to commutative orthogonal reflection field $T\omega$. The first result of this section is the following characterization of the sphere $S^m\left(c\right)$:

Theorem 1.

An m-dimensional compact and connected Riemannian manifold $(M^m,g)$, $m>2$, with scalar curvature τ, admits a nontrivial CONFLVF ω with conformal potential θ, associated tensor T satisfying $T\omega =0$ and the proportionality function ρ satisfying

$$\rho =-{\displaystyle \frac{\tau }{m(m-1)}},$$

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

Proof. 

Suppose an m-dimensional compact and connected Riemannian manifold $(M^m,g)$, $m>2$, with scalar curvature $\tau$, admits a nontrivial CONFLVF $\omega$ with conformal potential $\theta$, associated tensor T satisfying

$$T\omega =0$$

(27)

and the proportionality function $\rho$ satisfying

$$\rho =-{\displaystyle \frac{\tau }{m(m-1)}}.$$

(28)

Then, by Corollary 1, we have

$$\nabla \theta =-{\displaystyle \frac{\tau }{m(m-1)}}\omega ,$$

(29)

which on differentiation with respect to a smooth vector field X on $M^m$ and using Equation (9) provides

$${\mathcal{H}}^{\theta }X=-{\displaystyle \frac{1}{m(m-1)}}X\left(\tau \right)\omega -{\displaystyle \frac{\tau }{m(m-1)}}\theta X-{\displaystyle \frac{\tau }{m(m-1)}}TX.$$

Rewriting above equation in the form

$$\left({\mathcal{H}}^{\theta }+{\displaystyle \frac{\tau }{m(m-1)}}\theta I\right)\left(X\right)=-{\displaystyle \frac{1}{m(m-1)}}X\left(\tau \right)\omega -{\displaystyle \frac{\tau }{m(m-1)}}TX$$

(30)

and noticing that the left hand is symmetric, and using the skew symmetry of the associated operator T, we conclude

$$X\left(\tau \right)g\left(\omega ,Y\right)-Y\left(\tau \right)g\left(\omega ,X\right)+2\tau g\left(TX,Y\right)=0,$$

which confirms that

$$2\tau TX=g\left(\omega ,X\right)\nabla \tau -X\left(\tau \right)\omega .$$

(31)

Taking a local frame $\left\{v_1,..,v_m\right\}$ and using it with above equation, we compute

$$4{\tau }^2{∥T∥}^2=\sum _{k}g\left(2\tau T{v}_k,2\tau T{v}_k\right)=\sum _{k}g\left(g\left(\omega ,v_k\right)\nabla \tau -v_k\left(\tau \right)\omega ,g\left(\omega ,v_k\right)\nabla \tau -v_k\left(\tau \right)\omega \right),$$

that is,

$$2{\tau }^2{∥T∥}^2={∥\nabla \tau ∥}^2{∥\omega ∥}^2-\omega {\left(\tau \right)}^2.$$

(32)

Now, taking $X=\omega$ in Equation (31) and using Equation (27), we conclude

$${∥\omega ∥}^2\nabla \tau =\omega \left(\tau \right)\omega$$

(33)

and taking the inner product with $\nabla \tau$ in above equation confirms

$${∥\nabla \tau ∥}^2{∥\omega ∥}^2=\omega {\left(\tau \right)}^2.$$

Using above equation in Equation (32), we obtain ${\tau }^2{∥T∥}^2=0$. If $\tau =0$, then Equation (29) will imply that the conformal potential $\theta$ is a constant and using it with the integral of Equation (15) would yield $\theta =0$, which is contrary to the assumption that $\omega$ is a nontrivial CONFLVF. Hence, we must have associated tensor $T=0$ and its effect on Equation (13) is

$$S\omega =-(m-1)\nabla \theta .$$

Inserting Equation (29) in above equation, it changes to

$$S\omega ={\displaystyle \frac{\tau }{m}}\omega .$$

Differentiating this relation with respect to a smooth vector field X on $M^m$ while using Equation (9) and $T=0$, we arrive at

$$\left({\nabla }_{X}S\right)\left(\omega \right)+S\left(\theta X\right)={\displaystyle \frac{1}{m}}\left(X\left(\tau \right)\omega +\tau \theta X\right).$$

Using $X=v_k$ in above equation and then taking the inner product with $v_k$ and summing the resultant equation over a local frame $\left\{v_1,\cdots ,v_k\right\}$, while using the symmetry of the Ricci operator S and Equation (14), we conclude

$${\displaystyle \frac{1}{2}}\omega \left(\tau \right)+\theta \tau ={\displaystyle \frac{1}{m}}\omega \left(\tau \right)+\theta \tau ,$$

that is,

$${\displaystyle \frac{m-2}{2m}}\omega \left(\tau \right)=0.$$

Since, $m>2$, we have $\omega \left(\tau \right)=0$ and combining it with Equation (33) and the fact that $\omega$ is a nontrivial CONFLVF, we conclude that $\tau$ is a constant. Thus, by Equation (28) the proportionality function $\rho$ is a constant. Hence, by Proposition 1, we see that $\left(M^m,g\right)$ is isometric to $S^n\left(c\right)$, where $\tau =m(m-1)c$. The converse is trivial and immediately follows through Equation (3). □

Next, we obtain the following characterization of the sphere $S^m\left(c\right)$:

Theorem 2.

An m-dimensional compact and connected Riemannian manifold $\left(M^m,g\right)$, $m>1$, admits a nontrivial CONFLVF ω with conformal potential θ, commutative orthogonal reflection $T\omega$, the proportionality function ρ constant along the integral curves of ω and the Ricci curvature $\mathrm{Ric}\left(\nabla \theta ,\nabla \theta \right)$ satisfying

$${\int }_{M^m}\mathrm{Ric}\left(\nabla \theta ,\nabla \theta \right)d{V}_g\ge m(m-1){\int }_{M^m}{\rho }^2{\theta }^{2}d{V}_g,$$

if and only if $\left(M^m,g\right)$ is isometric to $S^m\left(c\right)$.

Proof. 

Since, orthogonal reflection $T\omega$ is commutative, by Lemma 1, we have

$$\nabla \theta =\rho \omega ,$$

(34)

where the proportionality function $\rho$ is constant along the integral curves of $\omega$, that is

$$\omega \left(\rho \right)=0.$$

(35)

Differentiating Equation (34) while using Equation (9), we have

$${\mathcal{H}}^{\theta }X=X\left(\rho \right)\omega +\rho \theta X+\rho TX,$$

(36)

that is,

$$\left({\mathcal{H}}^{\theta }-\rho \theta I\right)X=X\left(\rho \right)\omega +\rho TX.$$

The left hand side of above equation is symmetric, using this information as well as that T is skew-symmetric through above equation, we obtain

$$X\left(\rho \right)g\left(\omega ,Y\right)-Y\left(\rho \right)g\left(\omega ,X\right)+2\rho g\left(TX,Y\right)=0,$$

which yields

$$2\rho TX=g\left(\omega ,X\right)\nabla \rho -X\left(\rho \right)\omega .$$

Inserting the above equation into Equation (36) gives

$${\mathcal{H}}^{\theta }X=\rho \theta X+{\displaystyle \frac{1}{2}}X\left(\rho \right)\omega +{\displaystyle \frac{1}{2}}g\left(\omega ,X\right)\nabla \rho .$$

(37)

We use a local frame $\left\{v_1,..,v_m\right\}$ and Equations (35) and (37) in computing

$${∥{\mathcal{H}}^{\theta }∥}^2=\sum _{k}g\left({\mathcal{H}}^{\theta }{v}_k,{\mathcal{H}}^{\theta }{v}_k\right)$$

and obtain

$${∥{\mathcal{H}}^{\theta }∥}^2={\displaystyle \frac{1}{2}}{∥\nabla \rho ∥}^2{∥\omega ∥}^2+m{\rho }^2{\theta }^2.$$

(38)

Taking divergence on both sides in Equation (34), while using Equations (15) and (35), we arrive at

$$\Delta \theta =m\rho \theta .$$

(39)

Now, inserting Equations (38) and (39) in Bochner’s Formula (24), we conclude

$${\int }_{M^m}\mathrm{Ric}\left(\nabla \theta ,\nabla \theta \right)d{V}_g={\int }_{M^m}\left(m^2{\rho }^2{\theta }^2-{\displaystyle \frac{1}{2}}{∥\nabla \rho ∥}^2{∥\omega ∥}^2-m{\rho }^2{\theta }^2\right)d{V}_g,$$

that is,

$$m(m-1){\int }_{M^m}{\rho }^2{\theta }^{2}d{V}_g-{\int }_{M^m}\mathrm{Ric}\left(\nabla \theta ,\nabla \theta \right)d{V}_g={\int }_{M^m}{\displaystyle \frac{1}{2}}{∥\nabla \rho ∥}^2{∥\omega ∥}^{2}d{V}_g.$$

On using the condition in the statement, we obtain

$${∥\nabla \rho ∥}^2{∥\omega ∥}^2=0.$$

However, $\omega$ being a nontrivial CONFLVF, the above equation implies that the proportionality function $\rho$ is a constant. Hence, by Proposition 1, we see that $\left(M^m,g\right)$ is isometric to $S^n\left(c\right)$, where $\rho =-c$ and $c>0$. Conversely, suppose $\left(M^m,g\right)$ is isometric to $S^n\left(c\right)$. Then, by Equation (3), we see that $S^m\left(c\right)$ admits a nontrivial CONFLVF $\omega$ with conformal potential $\theta =-\sqrt{c}f$ and that $\omega$ being closed its associated tensor $T=0$ and therefore, $T\omega$ is automatically commutative with proportionality function $\rho =-c$ (seen through $\nabla \theta =-\sqrt{c}\nabla f=-c\omega$). Thus, we confirm that the proportionality function is constant along the integral curves of the CONFLVF $\omega$. Finally, through Equation (3), we have $\Delta \theta =-\sqrt{c}\Delta f=-c\mathrm{div}\omega =-mc\theta$, that is, $\theta \Delta \theta =-mc{\theta }^2$ and integrating last relation, yields

$${\int }_{S^m\left(c\right)}{∥\nabla \theta ∥}^{2}d{V}_g=mc{\int }_{S^m\left(c\right)}{\theta }^{2}d{V}_g$$

(40)

Now, the Ricci curvature $R\left(\nabla \theta ,\nabla \theta \right)$ for the sphere $S^m\left(c\right)$ is given by $\mathrm{Ric}\left(\nabla \theta ,\nabla \theta \right)=(m-1)c{∥\nabla \theta ∥}^2$, which in view of Equation (40) and $\rho =-c$ implies

$${\int }_{S^m\left(c\right)}\mathrm{Ric}\left(\nabla \theta ,\nabla \theta \right)d{V}_g=m(m-1){\int }_{S^m\left(c\right)}{\rho }^2{\theta }^{2}d{V}_g.$$

Hence, the converse holds. □

Finally, in this section, we obtain the following characterization of the sphere $S^m\left(c\right)$:

Theorem 3.

Let ω be a nontrivial CONFLVF on an m-dimensional compact and connected Riemannian manifold $\left(M^m,g\right)$, $m>1$, with conformal potential θ, associated operator T with scalar curvature τ constant along the integral curves of ω. If the Ricci operator S satisfies $S\left(\omega \right)=\lambda \omega$ for a nonzero constant λ and orthogonal reflection field $T\omega =0$, then τ is a positive constant and $\left(M^m,g\right)$ is isometric to $S^m\left(c\right)$, $\tau =m(m-1)c$. Moreover, the converse is also true.

Proof. 

Since,

$$T\omega =0$$

(41)

by Corollary 1, we have

$$\nabla \theta =\rho \omega .$$

(42)

Also, we have

$$S\left(\omega \right)=\lambda \omega \mathrm{and}\omega \left(\tau \right)=0,$$

(43)

where $\lambda$ is a nonzero constant. Multiplying first equation in (43) by proportionality function $\rho$ and using Equation (42), we have

$$\lambda \nabla \theta =S\left(\nabla \theta \right).$$

Differentiating above equation with respect to a smooth vector field X on $M^m$, we arrive at

$$\lambda {\mathcal{H}}^{\theta }X=\left({\nabla }_{X}S\right)\left(\nabla \theta \right)+S\left({\mathcal{H}}^{\theta }X\right).$$

(44)

However, using Equations (9) and (42), we obtain

$${\mathcal{H}}^{\theta }X=X\left(\rho \right)\omega +\rho \theta X+\rho TX.$$

Using above relation and Equation (43) in (44), it changes to

$$\lambda {\mathcal{H}}^{\theta }X=\left({\nabla }_{X}S\right)\left(\nabla \theta \right)+\lambda X\left(\rho \right)\omega +\rho \theta S\left(X\right)+\rho S\left(TX\right).$$

Taking trace in above equation while noticing that $Tr.\left(S\circ T\right)=0$ owing to symmetry of S and skew symmetry of T, we conclude

$$\lambda \Delta \theta ={\displaystyle \frac{1}{2}}g\left(\nabla \theta ,\nabla \tau \right)+\lambda \omega \left(\rho \right)+\rho \theta \tau .$$

Note that by Equations (42) and the second equation in (43), we see that $g\left(\nabla \theta ,\nabla \tau \right)=0$. Therefore, the above equation is now

$$\lambda \Delta \theta =\lambda \omega \left(\rho \right)+\rho \theta \tau .$$

(45)

Now, taking divergence on both sides of Equation (43), and using Equation (15), we reach

$$\Delta \theta =\omega \left(\rho \right)+m\rho \theta ,$$

which, on insertion in Equation (45), reduces it to

$$\rho \theta \left(\tau -m\lambda \right)=0.$$

(46)

If $\rho =0$, then Equation (42) implies that $\theta$ is a constant, which by integration of Equation (15) implies constant $\theta =0$, a contradiction to the fact that $\omega$ is a nontrivial CONFLVF. With same reasoning, we have $\rho \theta \ne 0$. Thus, Equation (46) confirms that $\tau =m\lambda$ is a constant. On using Equations (41) and (42), we have

$$T\left(\nabla \theta \right)=0.$$

Differentiating above relation with respect to a smooth vector field X on $M^m$, we obtain

$$\left({\nabla }_{X}T\right)\left(\nabla \theta \right)+T\left({\mathcal{H}}^{\theta }X\right)=0.$$

Using Equation (16) in above equation, it changes to

$$R\left(X,\omega \right)\nabla \theta +\nabla \theta \left(\theta \right)X-g\left(X,\nabla \theta \right)\nabla \theta +T\left({\mathcal{H}}^{\theta }X\right)=0$$

and taking trace in this equation and noticing that $Tr.\left(T\circ {\mathcal{H}}^{\theta }\right)=0$, we arrive at

$$\mathrm{Ric}\left(\omega ,\nabla \theta \right)+\left(m-1\right){∥\nabla \theta ∥}^2=0.$$

Multiplying the above equation by $\rho$ and using Equation (42), we confirm

$$\mathrm{Ric}\left(\nabla \theta ,\nabla \theta \right)+\left(m-1\right)\rho {∥\nabla \theta ∥}^2=0.$$

(47)

However, the first equation in (43) on multiplication by $\rho$ provides $S\left(\nabla \theta \right)=\lambda \nabla \theta$, that is, $\mathrm{Ric}\left(\nabla \theta ,\nabla \theta \right)=\lambda {∥\nabla \theta ∥}^2$ and combining this last relation with Equation (47), we obtain

$$\left(\lambda +(m-1)\rho \right){∥\nabla \theta ∥}^2=0.$$

Since, $\omega$ is a nontrivial CONFLVF, $\theta$ is non-constant as argued earlier. Thus, the above equation and $m>1$ imply the proportionality function

$$\rho =-{\displaystyle \frac{\lambda }{m-1}}.$$

Thus, the proportionality function $\rho$ is a constant and as $\tau =m\lambda$, we have $\tau =-m(m-1)\rho$. Hence, by Proposition 1, $\left(M^m,g\right)$ is isometric to the sphere $S^m\left(c\right)$. The converse is trivial. □

4. Characterizations of Euclidean Spaces

On the Euclidean space $\left(R^m,〈,〉\right)$ there are many nontrivial conformal vector fields. For instance, the position vector field $\xi$

$$\xi =\sum _j{u}^j{\displaystyle \frac{\partial }{\partial u^j}}$$

(48)

satisfies

$${\displaystyle \frac{1}{2}}{\pounds }_{\xi }g=g,$$

where $g=〈,〉$ is the Euclidean metric, that is, $\xi$ is a CONFLVF on $\left(E^n,〈,〉\right)$ with conformal potential $\theta =1$. However, $\xi$ is closed and therefore its associated tensor $T=0$. Next, we construct a non-closed nontrivial CONFLVF on $\left(E^n,〈,〉\right)$. Define a vector field $\varsigma$ on $R^m$, $m>2$, by

$$\varsigma =\xi +u^2{\displaystyle \frac{\partial }{\partial u^1}}-u^1{\displaystyle \frac{\partial }{\partial u^2}},$$

Then, we see that

$${\nabla }_X\varsigma =X+TX,$$

(49)

where

$$TX=X\left(u^2\right){\displaystyle \frac{\partial }{\partial u^1}}-X\left(u^1\right){\displaystyle \frac{\partial }{\partial u^2}}$$

(50)

and it follows that $〈TX,Y〉=-〈X,TY〉$, that is, T is a skew-symmetric $(1,1)$ tensor on the Euclidean space $\left(R^m,〈,〉\right)$. On using Equation (49), we see that

$${\displaystyle \frac{1}{2}}{\pounds }_{\varsigma }g=g,$$

that is, $\varsigma$ is a nontrivial CONFLVF on $\left(R^m,〈,〉\right)$ with conformal potential $\theta =1$, associated tensor T given by Equation (50). Moreover, $\varsigma$ is not a closed CONFLVF. Moreover, we see that there are many of these type of nontrivial non-closed conformal vector fields on the Euclidean space $\left(R^m,〈,〉\right)$.

In this section, we find the following characterizations for a Euclidean space.

Theorem 4.

Let ω be a nontrivial CONFLVF on an m-dimensional complete and connected Riemannian manifold $\left(M^m,g\right)$, $m>1$, with conformal potential θ, associated operator T with scalar curvature τ. Then, orthogonal reflection $T\omega =0$ and the proportionality function ρ satisfies

$$\rho \mathrm{Ric}\left(\omega ,\omega \right)\ge 0,$$

if and only if $\left(M^m,g\right)$ is isometric to the Euclidean space $\left(R^m,〈,〉\right)$.

Proof. 

Suppose the associated tensor T satisfies

$$T\omega =0.$$

(51)

Then, by Corollary 1, we have

$$\nabla \theta =\rho \omega .$$

(52)

Multiplying Equation (51) by the proportionality function $\rho$ and using Equation (52), we obtain

$$T\left(\nabla \theta \right)=0,$$

which, on differentiating the above equation with respect to a smooth vector field X, gives

$$\left({\nabla }_{X}T\right)\left(\nabla \theta \right)+T\left({\mathcal{H}}^{\theta }X\right)=0.$$

Treating above equation with Equation (16), we obtain

$$R\left(X,\omega \right)\nabla \theta +\nabla \theta \left(\theta \right)X-g\left(X,\nabla \theta \right)\nabla \theta +T\left({\mathcal{H}}^{\theta }X\right)=0.$$

On taking trace in above equation and noticing that $Tr.\left(T\circ {\mathcal{H}}^{\theta }\right)=0$, we conclude

$$\mathrm{Ric}\left(\omega ,\nabla \theta \right)+(m-1){∥\nabla \theta ∥}^2=0,$$

which in view of Equation (52) implies

$$\rho \mathrm{Ric}\left(\omega ,\omega \right)+(m-1){∥\nabla \theta ∥}^2=0.$$

Now, using the condition in the statement, the above equation yields $\nabla \theta =0$, that is, the conformal potential $\theta$ is a constant, say c. As $\omega$ nontrivial CONFLVF, we have constant $c\ne 0$. Define a function

$$\phi ={\displaystyle \frac{1}{2}}{∥\omega ∥}^2$$

and use Equations (9) and (51) in order to compute the gradient $\nabla \phi$ of $\phi$ to be given by

$$\nabla \phi =c\omega .$$

(53)

Differentiate the above equation with respect to a smooth vector field X on $M^m$, and using Equation (9), we obtain

$${\mathcal{H}}^{\phi }X=c^{2}X+cTX,$$

which on using symmetry and skew symmetry arguments yields ${\mathcal{H}}^{\phi }X=c^{2}X$ and $cTX=0$. We have arrived at

$$\mathrm{Hess}\left(\phi \right)=c^{2}g,$$

(54)

where Equation (53) forbids $\phi$ to be a constant due to the fact that $c\ne 0$ and that $\omega$ is nontrivial CONFLVF. Equation (54) confirms that $\left(M^m,g\right)$ is isometric to the Euclidean space $\left(R^m,〈,〉\right)$ (cf. [4]). The converse is trivial as the position field $\xi$ in Equation (48) is a nontrivial CONFLVF with conformal potential $\theta =1$ and is a closed vector field of the associated tensor $T=0$, and as $\left(R^m,〈,〉\right)$ is flat, the condition $\rho \mathrm{Ric}\left(\xi ,\xi \right)=0$. □

Theorem 5.

An m-dimensional complete and connected Riemannian manifold $\left(M^m,g\right)$, $m>1$, admits a nontrivial CONFLVF ω with conformal potential θ constant along the integral curves of ω, orthogonal reflection $T\omega =0$ if and only if $\left(M^m,g\right)$ is isometric to the Euclidean space $\left(R^m,〈,〉\right)$.

Proof. 

Suppose $T\omega =0$ holds, which with the help of Corollary 1 makes it possible that $\nabla \theta =\rho \omega$. Taking the inner product in the last relation by $\nabla \theta$ and using the statement that $\theta$ constant along the integral curves of $\omega$, we conclude

$${∥\nabla \theta ∥}^2=0,$$

that is, the conformal potential $\theta$ is a constant, say c, and this constant $c\ne 0$ due to the fact that $\omega$ is a nontrivial CONFLVF. The rest of the proof is similar to that of Theorem 4. □

5. An Additional Result

In this section, we obtain the following integral condition for a CONFLVF $\omega$ on a compact Riemannian manifold $\left(M^m,g\right)$ to be closed.

Theorem 6.

Let ω be a CONFLVF on an m-dimensional compact Riemannian manifold $\left(M^m,g\right)$. Then

$${\int }_{M^m}\left[\mathrm{Ric}\left(\omega ,\omega \right)-{\displaystyle \frac{m-1}{m}}{\left(\mathrm{div}\omega \right)}^2\right]d{V}_g=0,$$

if and only if ω is closed.

Proof. 

We recall from (cf. [4], p. 46) that

$${\int }_{M^m}\left[\mathrm{Ric}\left(\omega ,\omega \right)-{∥\nabla \omega ∥}^2-{\displaystyle \frac{m-2}{m}}{\left(\mathrm{div}\omega \right)}^2\right]d{V}_g=0$$

(55)

for a CONFLVF $\omega$ on $\left(M^m,g\right)$. Now, using Equation (9), and skew symmetry of T, we have

$${∥\nabla \omega ∥}^2=m{\theta }^2+{∥T∥}^2.$$

(56)

Substituting the value of ${∥\nabla \omega ∥}^2$ from (56) in (55), and using Equation (15), gives

$${\int }_{M^m}\left[\mathrm{Ric}\left(\omega ,\omega \right)-m{\theta }^2-{∥T∥}^2-{\displaystyle \frac{m-2}{m}}{\left(m\theta \right)}^2\right]d{V}_g=0,$$

that is,

$${\int }_{M^m}\left[\mathrm{Ric}\left(\omega ,\omega \right)-m(m-1){\theta }^2-{∥T∥}^2\right]d{V}_g=0.$$

(57)

Here, we would like to point out that above integral formula is also obtained in [15]. Using Equation (15), the above equation becomes

$${\int }_{M^m}\left[\mathrm{Ric}\left(\omega ,\omega \right)-{\displaystyle \frac{m-1}{m}}{\left(\mathrm{div}\omega \right)}^2-{∥T∥}^2\right]d{V}_g=0,$$

which shows that

$${\int }_{M^m}\left[\mathrm{Ric}\left(\omega ,\omega \right)-{\displaystyle \frac{m-1}{m}}{\left(\mathrm{div}\omega \right)}^2\right]d{V}_g\ge 0$$

(58)

with equality holds if and only if $T=0$, that is, $\omega$ is closed. □

Let us see whether the equality case ($\omega$ closed) holds on the unit sphere $S^m$ for $\omega =-\nabla \theta$. For $S^m$, $\Delta \theta =-\mathrm{div}\omega =-m\theta$ and $T=0$. Then Equation (57) is

$${\int }_{S^m}\left[\mathrm{Ric}\left(\omega ,\omega \right)-m(m-1){\theta }^2\right]d{V}_g=0.$$

The above equation in view of $\mathrm{Ric}\left(\omega ,\omega \right)=(m-1){∥\omega ∥}^2=(m-1){∥\nabla \theta ∥}^2$ changes to

$${\int }_{S^m}\left[{∥\nabla \theta ∥}^2-m{\theta }^2\right]d{V}_g=0,$$

which is true, as on $S^m$, $\theta \Delta \theta =-m{\theta }^2$, integrates to

$${\int }_{S^m}{∥\nabla \theta ∥}^{2}d{V}_g={\int }_{S^m}m{\theta }^{2}d{V}_g.$$