A remarkable property of concircular vector fields on a Riemannian manifold

Abstract

In this paper, we show that given a nontrivial concircular vector field $\mathbf{u}$ on a Riemannian manifold $(M,g)$ with potential function $f$, there exists a unique smooth function $\rho$ on $M$ that connects $\mathbf{u}$ to the gradient of potential function $\nabla f$, which we call the connecting function of the concircular vector field $\mathbf{u}$. 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.

10 pages