Characterizations of Einstein manifolds and odd-dimensional spheres

In this paper, we use a Killing vector field on a Riemannian manifold to characterize odd-dimensional spheres and Einstein manifolds.

► An important question in geometry of a Riemannian manifold is to characterize Einstein manifolds.
► We use a Killing vector field to characterize Einstein manifolds.
► A necessary condition for a Riemannian manifold to be isometric to an Odd-dimensional sphere is obtained using a Killing vector field.
► A characterization of the Euclidean space is found using Killing vector fields.