标题： 某些具有共形向量场的黎曼流形的特征描述 显示英文标题

标题： Some characterizations of Riemannian manifolds endowed with a conformal vector fields

摘要： 本文的目的是研究紧致黎曼流形 $M$（带边或不带边 $\partial M$）上具有共形因子 $\rho$ 的共形向量场 $\xi$ 的存在性。我们首先证明：若紧致黎曼流形 $(M^n, g)\,,n \geq 3,$ 具有常数量曲率且边界 $\partial M$ 完全测地，并且迹零Ricci曲率在方向 $\nabla \rho,$ 上为零，则该流形与标准半球面等距。 在$4$维情况下，在条件$\displaystyle\int_M|\mathring{Ric}|^2\langle \xi,\nabla \rho\rangle \,dM\leq0$下，我们证明，或者$M$与标准球面等距，或者$M$与标准半球面等距。 最后，我们给出了宇宙无毛猜想的部分答案。 显示英文摘要

摘要： 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.