Conformal Vector Fields and the De-Rham Laplacian on a Riemannian Manifold with Boundary

Abstract

Let (M n , g) be an n-dimensional compact connected Riemannian manifold with smooth boundary. In this article, we study the effects of the presence of a nontrivial conformal vector field on (M n , g). We used the wellknown de-Rham Laplace operator and a nontrivial solution of the famous Fischer-Marsden differential equation to provide two characterizations of the hemisphere S n + (c) of constant curvature c > 0. As a consequence of the characterization using the Fischer-Marsden equation, we prove the cosmic no-hair conjecture under a given integral condition.