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

Given a conformal vector field $ X $ defined on an $ n $-dimensional Riemannian manifold $ \left(N^{n}, g\right) $, naturally associated to $ X $ are the conformal factor $ \sigma $, a smooth function defined on $ N^{n} $, and a skew symmetric $ (1, 1) $ tensor field $ \Omega $, called the associated tensor, that is defined using the $ 1 $-form dual to $ X $. In this article, we prove two results. In the first result, we show that if an $ n $-dimensional compact and connected Riemannian manifold $ \left(N^{n}, g\right) $, $ n > 1 $, of positive Ricci curvature admits a nontrivial (non-Killing) conformal vector field $ X $ with conformal factor $ \sigma $ such that its Ricci operator $ Rc $ and scalar curvature $ \tau $ satisfy

$ Rc\left( X\right) = -(n-1)\nabla \sigma \; \; \text{ and }\; \; X(\tau ) = 2\sigma \left( n(n-1)c-\tau \right) $

for a constant $ c $, necessarily $ c > 0 $ and $ \left(N^{n}, g\right) $ is isometric to the sphere $ S_{c}^{n} $ of constant curvature $ c $. The converse is also shown to be true. In the second result, it is shown that an $ n $-dimensional complete and connected Riemannian manifold $ \left(N^{n}, g\right) $, $ n > 1 $, admits a nontrivial conformal vector field $ X $ with conformal factor $ \sigma $ and associated tensor $ \Omega $ satisfying

$ Rc\left( X\right) = -div\Omega \; \; \text{ and }\; \; \Omega \left( X\right) = 0, $

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