  Let $(M^n,g)$ be an $n$-dimensional compact connected Riemannian manifold
with boundary. In this article, we study the effects of the presence of a
nontrivial conformal vector field on $(M^n,g)$. We used the wekk-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.
