Ricci Vector Fields Revisited

We continue studying the σ -Ricci vector field u on a Riemannian manifold ( N m , g ) , which is not necessarily closed. A Riemannian manifold with Ricci operator T , a Coddazi-type tensor, is called a T - manifold . In the first result of this paper, we show that a complete and simply connected T - manifold ( N m , g ) , m > 1 , of positive scalar curvature τ , admits a closed σ -Ricci vector field u such that the vector u − ∇ σ is an eigenvector of T with eigenvalue τ m − 1 , if and only if it is isometric to the m -sphere S α m . In the second result, we show that if a compact and connected T - manifold ( N m , g ) , m > 2 , admits a σ -Ricci vector field u with σ ≠ 0 and is an eigenvector of a rough Laplace operator with the integral of the Ricci curvature R i c u , u that has a suitable lower bound, then ( N m , g ) is isometric to the m -sphere S α m , and the converse also holds. Finally, we show that a compact and connected Riemannian manifold ( N m , g ) admits a σ -Ricci vector field u with σ as a nontrivial solution of the static perfect fluid equation, and the integral of the Ricci curvature R i c u , u has a lower bound depending on a positive constant α , if and only if ( N m , g ) is isometric to the m -sphere S α m .