Some new characterizations of spheres and Euclidean spaces using conformal vector fields

1.   Introduction

Conformal geometry is one of the oldest branches of differential geometry, as one can follow it through ^[1] as old as 1959. It is evolving with time and getting enriched constantly till one can find the most recent work in ^[2]. The main topic in the conformal geometry is about studying the influence of a conformal vector field $X$ on an $n$-dimensional Riemannian manifold $\left(N^{n}, g\right)$. We shall abbreviate a conformal vector field $X$ by CONFVF $X$ for the sake of convenience. There is a smooth function $\sigma$ that is naturally associated to a CONFVF$X$ on $\left(N^{n}, g\right)$ called the conformal factor satisfying

where $£$ is the Lie derivative operator. A CONFVF $X$ is said to be Killing if the conformal factor $\sigma = 0$, and consequently, a nontrivial CONFVF $X$ must have a conformal factor $\sigma \neq 0$. There is naturally associated a $(1, 1)$ skew symmetric tensor field $\Omega$ to a CONFVF $X$ on $\left(N^{n}, g\right)$ called the associated tensor of CONFVF $X$, defined by

for smooth vector fields $E, F$ on $N^{n}$, where $\eta$ is the $1$-form dual to $X$. This associated tensor $\Omega$ plays a crucial role in studying the impact of a CONFVF $X$ on the geometry of $\left(N^{n}, g\right)$ (see ^[2]).

The sphere $S_{c}^{n}$ of constant curvature as a hypersurface of the Euclidean space $\left(E^{n+1}, \langle, \rangle \right)$, where $\langle, \rangle$ is the Euclidean metric, has unit normal $\zeta$, induced metric $g$, and the Weingarten map

Choosing a unit constant vector field $Z$ on the Euclidean space $\left(E^{n+1}, \langle, \rangle \right)$, its tangential component $X$ to the sphere $S_{c}^{n}$ satisfies

where

and $\nabla \rho$ is the gradient of $\rho$ on $S_{c}^{n}$ with respect to the induced metric $g$. Thus, we see that $X$ is a CONFVF on the sphere $S_{c}^{n}$ with conformal factor

This CONFVF $X$ is closed, and therefore the associated tensor $\Omega = 0$.

Also, using complex structure $J$ on $\left(E^{2n}, \langle, \rangle \right)$, define a unit vector field $\xi = J\zeta$, which has covariant derivative

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

Now, define a vector field

we obtain a CONFVF$\overline{X}$ with conformal factor $\sigma$ that is not closed and indeed has an associated tensor

Non-closed CONFVF are in abundance, for instance on the Euclidean space $\left(E^{2n}, \langle, \rangle \right)$, if $\xi$ is the position vector field on $E^{2n}$, then

where $J$ is the complex structure, is a CONFVF on $\left(E^{2n}, \langle, \rangle \right)$ that is not closed and has an associated tensor

Riemannian manifolds that admit closed CONFVF have been extensively studied ^[3,4,5]. For related work on conformal vector fields that are not necessarily closed, refer to ^[6,7,8,9]. We see that while studying the impact of a CONFVF $X$ on the geometry of the Riemannian manifold $\left(N^{n}, g\right)$, on which it is defined, the associated tensor $\Omega$ offers some resistance in analyzing the geometry. Therefore, this study becomes smooth once one assumes that the CONFVF $X$ is closed, which forces the associated tensor $\Omega$ to vanish. The study of the impact of the existence of a non-closed CONFVF on Riemannian manifolds is relatively difficult, and therefore this difficulty is softened by imposing geometric restrictions on Riemannian manifolds, such as the scalar curvature $\tau$ being constant. Riemannian manifolds admitting non-closed CONFVF have been studied in ^[8,10,11]. Moreover, apart from the fact that the presence of a CONFVF on a Riemannian manifold influences its geometry, they are also used in the theory of relativity ^[12,13,14].

Observe that the Ricci operator $Rc$ on a Riemannian manifold $(N^{n}, g)$ is related to the Ricci tensor $Ric$ by

and for the sphere $S_{c}^{n}$ the Ricci operator is given by

Moreover, the conformal factor $\sigma$ of the CONFVF $X$ on $S_{c}^{n}$ described in Eq (1.3) satisfies

It naturally raises a question: Under what conditions is a compact and connected Riemannian manifold $(N^{n}, g)$ admitting a nontrivial CONFVF$X$ with conformal factor $\sigma$ satisfying Eq (1.5) is isometric to the sphere $S_{c}^{n}$?

The reader may refer to the following sources, as well as the references in ^{[10,15,16,17]}, for additional information on this question. In this article, we answer this question and indeed find a new characterization of the sphere $S_{c}^{n}$ (see Theorem 1). Finally, in this paper, we find a characterization of the Euclidean space $\left(E^{n}, \langle, \rangle \right)$ using a nontrivial CONFVF $X$ with conformal factor $\sigma$ on a complete and connected Riemannian manifold $(N^{n}, g)$ (see Theorem 2).

2.   Preliminaries

Let $X$ be a nontrivial CONFVF on an $n$-dimensional Riemannian manifold $(N^{n}, g)$ with conformal factor $\sigma$. Then employing Eqs (1.1) and (1.2) in Koszul's formula, we have

where $\nabla _{E}$ is the covariant derivative with respect to the Riemannian connection on $(N^{n}, g)$ and $E, F$ are smooth vector fields on $N^{n}$. Consequently, we have

where $\Omega$ is the associated tensor associated with CONFVF $X$.

Using Eq (2.1), we find the following expression for the curvature tensor field $R$

for smooth vector fields $E, F$ on $N^{n}$. Choosing a local frame $\left\{ E_{1}, \dots, E_{n}\right\}$ on $(N^{n}, g)$ in the above equation in order to compute the Ricci tensor $Ric$, we obtain

where

Thus, for the CONFVF $X$ on an $n$-dimensional Riemannian manifold $(N^{n}, g)$ with conformal factor $\sigma$, on using Eq (2.3), for Ricci operator $Rc$, we have

On a Riemannian manifold $(N^{n}, g)$, the scalar curvature $\tau = Tr.Rc$ satisfies the following ^[18,19]

where

We see that for the CONFVF $X$ on an $n$-dimensional Riemannian manifold $(N^{n}, g)$ with conformal factor $\sigma$, on using Eq (2.1), we have

Also, we compute the divergence of the vector field $Rc\left(X\right)$ as follows:

which, on using the symmetry of $Rc$ and Eqs (2.1) and (2.5), yields

and $\Omega$ being skew symmetric, we reach at

Similarly, on using Eq (2.1) and the skew symmetry of the associated operator $\Omega$ for the CONFVF $X$ on an $n$-dimensional Riemannian manifold $(N^{n}, g)$ with conformal factor $\sigma$, we find

where

Now, for the conformal factor $\sigma$ of the CONFVF $X$ on an $n$ -dimensional Riemannian manifold $(N^{n}, g)$, the Hessian $Hess(\sigma)$ of $\sigma$ is the symmetric bilinear form defined by

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

for smooth vector fields $E, F$ on $N^{n}$. The Laplacian $\Delta \sigma$ of $\sigma$ is defined by

which is also given by

If $(N^{n}, g)$ is a compact Riemannian manifold, then we have the following Bochner's formula

where for a local frame $\left\{ E_{1}, .., E_{n}\right\}$ on $N^{n}$

3.   Characterizing spheres using conformal vector fields

Let $X$ be a nontrivial CONFVF on an $n$-dimensional Riemannian manifold $(N^{n}, g)$ with conformal factor $\sigma$. In this section, we shall answer the questions raised in the introduction. Indeed, first we prove the following:

Theorem 1. An $n$-dimensional compact and connected Riemannian manifold $(N^{n}, g)$, $n > 1$, of positive Ricci curvature and scalar curvature $\tau$ admits a nontrivial CONFVF $X$ with conformal factor $\sigma$ satisfying

for a constant $c$, if and only if $c > 0$ and $(N^{n}, g)$ is isometric to the sphere $S_{c}^{n}$.

Proof. Suppose $X$ is a nontrivial CONFVF on an $n$-dimensional compact and connected Riemannian manifold $(N^{n}, g)$, $n > 1$, of positive Ricci curvature with conformal factor $\sigma$, which satisfies

and

where $c$ is a constant. Using Eqs (2.4) and (3.1), we obtain $div \Omega = 0$ and inserting it in Eq (2.8) yields

Now, in Eq (3.1), taking the inner product with $X$, provides

which, in light of Eq (2.6) used in the formula

takes the form

Integrating the above equation leads to

Again, using Eq (3.1), we immediately have

Next, taking divergence on both sides of Eq (3.1) and making use of Eq (2.7), we get

which, on treating with Eq (3.2), reduces to

Multiplying the above equation by $\sigma$ and then integrating leads to

Also, we have

Integrating the above equation and using Eqs (2.9), (3.4), and (3.5), we arrive at

which is rearranged as

Now, inserting Eqs (3.6) and (3.7) in the above equation reveals

Observe that the integrand in the second integral in Eq (3.8) is non-negative by Schwarz inequality, and the hypothesis requires that the Ricci curvatures of the Riemannian manifold $\left(N^{n}, g\right)$ are positive. This traps Eq (3.8) to come forward with only the following solutions:

Interestingly, both equations in Eq (3.9) reach to the same conclusions. First, take the equation

which on differentiation with respect to a vector field $E$ on $N^{n}$ and using Eq (2.1), leads to the equation

where the left-hand side is symmetric and the right-hand side is skew symmetric. Hence, we have both

and

for arbitrary $E$. Thus, we have two choices: either $c = 0$ or $\Omega = 0$. If $c = 0$, Eq (3.6) will imply $\sigma$ is a constant, and then the integral of Eq (2.6) will produce $\sigma = 0$: and that is contrary to the fact that $X$ is nontrivial. Hence, $\Omega = 0$ and Eq (3.10) reduces to

where $\sigma$ has to be a non-constant function, for $\sigma$ a constant implies $\sigma = 0$, which is forbidden by $X$ being nontrivial. The second equation in Eq (3.9) also reaches the same conclusion as Eq (3.11). For it is the equality in Schwarz's inequality

which holds if and only if

and combining it with Eq (3.6), gives Eq (3.11). As seen in the above paragraph, the constant $c\neq 0$, indeed, as $\sigma$ is a non-constant function, Eq (3.7) reveals that $c > 0$. Thus, we have reached the conclusion that Eq (3.11) is Obata's differential equation ^[9,17]. Hence, $\left(N^{n}, g\right)$ is isometric to the sphere $S_{c}^{n}$.

Conversely, if $\left(N^{n}, g\right)$ is isometric to the sphere $S_{c}^{n}$, then through Eq (1.3), we have a CONFVF $X$ on $S_{c}^{n}$ with conformal factor

First, we claim that $X$ is a nontrivial CONFVF on $S_{c}^{n}$. For if $\sigma = 0$, that is, $\rho = 0$, which on using Eq (1.3) would imply $X = 0$, and in turn it would give

a contradiction to the fact that $Z$ is a unit vector. Hence, $X$ is a nontrivial CONFVF on $S_{c}^{n}$. Also, for $S_{c}^{n}$, we have

which, on treating with Eq (1.3), implies

that is, Eq (3.1) holds. Also, the scalar curvature $\tau$ of $S_{c}^{n}$ is

a constant, and therefore, Eq (3.2) holds. Finally, $S_{c}^{n}$ is compact and has positive Ricci curvature. Hence, converse is true. □

It is worth noticing that the condition (3.2) is essential in the above characterization of $S_{c}^{n}$, as there are compact manifolds admitting a nontrivial CONFVF that are not isometric to $S_{c}^{n}$ and on them the Eq (3.2) does not hold.

For example, consider the compact Riemannian manifold $\left(N^{n}, g\right)$, where

is the warped product, with $\rho$ a smooth positive function on the unit circle $S^{1}$ and the warped product metric

$\theta$ is a coordinate function on $S^{1}$ and $\overline{g}$ is the canonical metric on the sphere $S_{c}^{(n-1)}$ of constant curvature $c$. Then the vector field

on $\left(N^{n}, g\right)$ satisfies ^[19]

where $E$ is any vector field on $N^{n}$. Thus, we obtain

that is, $X$ is a CONFVF $\left(N^{n}, g\right)$ with conformal factor $\sigma = \rho ^{^{\prime }}$. We have the following expression for the Ricci operator $Rc$ on the warped product manifold $(N^{n}, g)$ ^[19]

for a horizontal vector field $E$ on $S^{1}$ and

for vertical vector field $V$ on $S_{c}^{n}$. As the CONFVF $X$ is horizontal, we see by Eq (3.12) that

that is, the condition (3.1) holds for the CONFVF $X$ on the compact warped product manifold $\left(N^{n}, g\right)$. The scalar curvature $\tau$ of $\left(N^{n}, g\right)$ is given by

which does not satisfy Eq (3.2).

4.   Characterizing Euclidean spaces by conformal vector fields

On the Euclidean space $\left(E^{n}, \langle, \rangle \right)$, there are finitely many nontrivial conformal vector fields. For instance, the position vector field $\xi$

satisfies

that is, $\xi$ is a CONFVF on $\left(E^{n}, \langle, \rangle \right)$ with conformal factor $\sigma = 1$. However, $\xi$ is closed, and therefore its associated tensor $\Omega = 0$. Next, we construct a non-closed nontrivial CONFVL on $\left(E^{n}, \langle, \rangle \right)$. Define a vector field $X$ on $E^{n}$, $n > 2$, by

Then, we see that

where

and it follows that

that is, $\Omega$ is a skew symmetric $(1, 1)$ tensor on the Euclidean space $\left(E^{n}, \langle, \rangle \right)$. Using Eq (4.2), one confirms that

that is, $X$ is a nontrivial CONFVF on $\left(E^{n}, \langle, \rangle \right)$ with conformal factor $\sigma = 1$, and it is not a closed vector field. Moreover, we see that there are finitely many of these types of nontrivial conformal vector fields on the Euclidean space $\left(E^{n}, \langle, \rangle \right)$.

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

Theorem 2. Let $X$ be a nontrivial CONFVF on an $n$-dimensional complete and connected Riemannian manifold $(N^{n}, g)$, $n > 1$, with conformal factor $\sigma$ and associated tensor $\Omega$. Then the following conditions hold:

if and only if $(N^{n}, g)$ is isometric to the Euclidean space $\left(E^{n}, \langle, \rangle \right)$.

Proof. Suppose an $n$-dimensional complete and connected Riemannian manifold $(N^{n}, g)$ admits a nontrivial CONFVL $X$ with conformal factor $\sigma$ and associated tensor $\Omega$ satisfying

and

Using Eqs (2.4), (4.3), and $n > 1$, we reach the conclusion that $\sigma$ is a constant. It is clear that the constant $\sigma \neq 0$ due to the fact that $X$ is nontrivial. Next, define a smooth function $\alpha$ by

Then, using Eqs (2.1) and (4.4), we find the gradient of the smooth function $\alpha$ is given by

Since the CONFVF $X$ is nontrivial and the constant $\sigma \neq 0$, the above equation confirms that the function $\alpha$ is not a constant. Now, differentiating Eq (4.5) with respect to a vector field $E$ on $N^{n}$ and using Eq (2.1), we arrive at

and taking the inner product in the above equation by $E$, yields

On polarizing the above equation, we conclude

where $\alpha$ is a non-constant function and $\sigma ^{2}$ is a nonzero constant. Equation (4.6) guarantees that $(N^{n}, g)$ is isometric to the Euclidean space $\left(E^{n}, \langle, \rangle \right)$.

The converse is trivial, for the position vector field $\xi$ given in (4.1) is a nontrivial CONFVF on the Euclidean space $\left(E^{n}, \langle, \rangle \right)$ with conformal factor $\sigma = 1$, and it being closed, we have associated tensor $\Omega = 0$. Also, for the Euclidean space $\left(E^{n}, \langle, \rangle \right)$ being flat, we have Ricci operator $Rc = 0$, and we see that the conditions (4.3) and (4.4) are automatically satisfied. Hence, the converse. □

5.   Conclusions

It is needless to mention that one of the most interesting and most sought-out questions in differential geometry is to find different characterizations of the model spaces, namely, the sphere $S_{c}^{n}$ of constant curvature $c$, the Euclidean space $E^{n}$, and the hyperbolic space $H_{-c}^{n}$ of constant curvature $-c$ ($c > 0$). In the present paper, we witnessed that an $n$-dimensional compact and connected Riemannian manifold $\left(N^{n}, g\right)$, $n > 1$, of positive Ricci curvature and scalar curvature admits a nontrivial CONFVF $X$ with conformal factor $\sigma$ satisfying the conditions

for a constant $c$, if and only if $c > 0$ and $\left(N^{n}, g\right)$ is isometric to the sphere $S_{c}^{n}$. In the process of the proof, we have seen that the above conditions together with the assumption that $\left(N^{n}, g\right)$ has positive Ricci curvature lead us to

Since $X$ is a CONFVF, this indicates that $\sigma$ is not constant, and furthermore, the compactness of $N^{n}$ forces the constant $c$ to be positive. Finally, combining Eq (5.2) with the definition of the CONFVF $X$ leads us to Obata's differential equation, concluding that $\left(N^{n}, g\right)$ is isometric to the sphere $S_{c}^{n}$.

Note that the CONFVF $X$ on the sphere $S_{c}^{n}$ with conformal factor $\sigma$, by virtue of Eq (1.3), satisfies

where

is the scalar curvature of the sphere $S_{c}^{n}$. This naturally leads to the question: Under what conditions a compact and connected Riemannian manifold $\left(N^{n}, g\right)$ with Ricci operator $Ric$ scalar curvature $\tau$, admitting a nontrivial CONFVF $X$ with conformal factor $\sigma$ satisfying Eq (5.3), that is

is isometric to the sphere $S_{c}^{n}$? We shall be interested in taking up this question in our future studies.

Author contributions

S. Deshmukh: conceptualization, investigation, methodology, validation, writing-original draft, formal analysis, validation, inspection; M. Guediri: conceptualization, investigation, methodology, formal analysis, writing-review and editing, validation, inspection. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

This research was supported by Researchers Supporting Project number (RSPD2024R1053), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare that there are no conflicts of interest.