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 $\mathrm{\Omega }$, 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

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

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 $\mathrm{\Omega }$ satisfying

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

if and only if $\left(N^n,g\right)$ is isometric to the Euclidean space $\left(E^n,⟨,⟩\right)$.