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 . [ABSTRACT FROM AUTHOR]