Some characterizations of Riemannian manifolds endowed with a conformal vector fields

TL;DR

This paper characterizes certain Riemannian manifolds with conformal vector fields, showing conditions under which they are isometric to standard spheres or hemispheres, and discusses implications for the cosmic no-hair conjecture.

Contribution

It provides new geometric characterizations of Riemannian manifolds with conformal vector fields, including conditions leading to standard sphere or hemisphere isometries, and offers partial insights into the cosmic no-hair conjecture.

Findings

Manifolds with constant scalar curvature and totally geodesic boundary are isometric to a hemisphere.

In 4D, under an integral condition, the manifold is either a sphere or a hemisphere.

Partial results are provided related to the cosmic no-hair conjecture.

Abstract

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 $\partial M$. We firstly prove that a compact Riemannian manifold $(M^n, g)\,,n \geq 3,$ with constant scalar curvature, with boundary $\partial M$ totally geodesic, in such way that the traceless Ricci curvature is zero in the direction of $\nabla \rho,$ is isometric to a standard hemisphere. In the $4$-dimensional case, under the condition $\displaystyle\int_M|\mathring{Ric}|^2\langle \xi,\nabla \rho\rangle \,dM\leq0$, 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.