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 well-known de-Rham Laplace operator and a nontrivial solution of the famous Fischer–Marsden differential equation to provide two characterizations of the hemisphere \({\mathbb {S}}^n_+(c)\) 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.