No abstract

Let (M n , g) be an n-dimensional compact connected Riemannian manifold with smooth boundary. We show that the presence of a nontrivial conformal gradient vector field on M , with an appropriate control on the Ricci curvature, causes M to be isometric to a hemisphere of S n . We also prove that if an Einstein manifold with boundary admits nonzero conformal gradient vector field, then its scalar curvature is positive and it is isometric to a hemisphere of S n . Furthermore, we prove that if M admits a nontrivial conformal vector field and has constant scalar curvature, then the scalar curvature is positive. Finally, a suitable control on the energy of a conformal vector field implies that M is isometric to a hemisphere S n + .

No abstract

No abstract

The aim of this paper is to prove that a compact almost Ricci soliton with null Cotton tensor is isometric to a standard sphere provided one of the following conditions associated to the Schouten tensor holds: the second symmetric function is constant and positive; two consecutive symmetric functions are non null multiple or some symmetric function is constant and the quoted tensor is positive.