Conformal gradient vector fields on Riemannian manifolds with boundary

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 + .

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Generalized Minkowski Type Integral Formulas for Compact Hypersurfaces in Pseudo-Riemannian Manifolds

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