Some characterizations of Riemannian manifolds endowed with a conformal vector fields

The aim of this article is to investigate the presence of a conformal vector $\xi$ with conformal factor $\rho$ on a compact Riemannian manifold $M$ with or without boundary $\mathrm{\partial }M$. We firstly prove that a compact Riemannian manifold $(M^n,g),n\ge 3,$ with constant scalar curvature, with boundary $\mathrm{\partial }M$ totally geodesic, in such way that the traceless Ricci curvature is zero in the direction of $\mathrm{\nabla }\rho ,$ is isometric to a standard hemisphere. In the $4$-dimensional case, under the condition ${\displaystyle {\int }_M\mathrm{\mid }\mathring{Ric}{\mathrm{\mid }}^2⟨\xi ,\mathrm{\nabla }\rho ⟩dM\le 0}$, we show that, either $M$ is isometric to a standard sphere, or $M$ is isometric to a standard hemisphere. Finally, we give a partial answer for the cosmic no-hair conjecture.