Homogeneous Riemannian manifolds of positive Ricci curvature

V. N. Berestovskii

Matematičeskie zametki, Tome 58 (1995) no. 3, pp. 334-340

Résumé

We prove that a homogeneous effective space $M=G/H$, where $G$ is a connected Lie group and $H\subset G$ is a compact subgroup, admits a $G$-invariant Riemannian metric of positive Ricci curvature if and only if the space $M$ is compact and its fundamental group $\pi_1(M)$ is finite (in this case any normal metric on $G/H$ is suitable). This is equivalent to the following conditions: the group $G$ is compact and the largest semisimple subgroup $LG\subset G$ is transitive on $G/H$. Furthermore, if $G$ is nonsemisimple, then there exists a $G$-invariant fibration of $M$ over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber.