A remarkable property of concircular vector fields on a Riemannian manifold

TL;DR

This paper investigates the properties of concircular vector fields on Riemannian manifolds, introducing a connecting function that links these fields to potential functions and characterizes spheres and Euclidean spaces.

Contribution

It introduces a unique connecting function for concircular vector fields and demonstrates its role in characterizing spheres, Euclidean spaces, and influencing manifold topology.

Findings

Existence of a unique connecting function for concircular vector fields.

Characterization of spheres and Euclidean spaces using the connecting function.

The connecting function affects the topology of the manifold.

Abstract

In this paper, we show that given a nontrivial concircular vector field $\boldsymbol{u}$ on a Riemannian manifold $(M,g)$ with potential function $f$, there exists a unique smooth function $\rho $ on $M$ that connects $\boldsymbol{u}$ to the gradient of potential function $\nabla f$, which we call the connecting function of the concircular vector field $\boldsymbol{u}$. Then this connecting function is shown to be a main ingredient in obtaining characterizations of $n$-sphere $\mathbf{S}^{n}(c)$ and the Euclidean space $\mathbf{E}^{n}$. We also show that the connecting function influences topology of the Riemannian manifold.