Spheres and Euclidean Spaces Via Concircular Vector Fields

Abstract

In this paper, we exhibit that non-trivial concircular vector fields play an important role in characterizing spheres, as well as Euclidean spaces. Given a non-trivial concircular vector field $\xi$ on a connected Riemannian manifold (M, g), two smooth functions $\sigma$ and $\rho$ called potential function and connecting function are naturally associated to $\xi$ . We use non-trivial concircular vector fields on n-dimensional compact Riemannian manifolds to find four different characterizations of spheres ${\mathbf {S}}^{n}(c)$. In particular, we prove an interesting result namely an n-dimensional compact Riemannian manifold (M, g) that admits a non-trivial concircular vector field $\xi$ such that the Ricci operator is invariant under the flow of $\xi$, if and only if, (M, g) is isometric to a sphere ${\mathbf {S}}^{n}(c)$. Similarly, we find two characterizations of Euclidean spaces ${\mathbf {E}}^{n}$. In particular, we show that an n-dimensional complete and connected Riemannian manifold (M, g) admits a non-trivial concircular vector field $\xi$ that annihilates the Ricci operator, if and only if, (M, g) is isometric to the Euclidean space ${\mathbf {E}}^{n}$.