Ricci Vector Fields Revisited

We continue studying the sigma-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 tau, admits a closed sigma-Ricci vector field u such that the vector u - del sigma is an eigenvector of T with eigenvalue tau m(-1), if and only if it is isometric to the m-sphere S-alpha(m). In the second result, we show that if a compact and connected T-manifold (N-m, g), m > 2, admits a sigma-Ricci vector field u with sigma not equal 0 and is an eigenvector of a rough Laplace operator with the integral of the Ricci curvature Ric(u, u) that has a suitable lower bound, then (N-m, g) is isometric to the m-sphere S-alpha(m), and the converse also holds. Finally, we show that a compact and connected Riemannian manifold (N-m, g) admits a sigma-Ricci vector field u with s as a nontrivial solution of the static perfect fluid equation, and the integral of the Ricci curvature Ric (u, u) has a lower bound depending on a positive constant a, if and only if (N-m, g) is isometric to the m-sphere S-alpha(m).