Invariant Einstein metrics on some homogeneous spaces of classical Lie groups

Andreas Arvanitoyeorgos; V. V. Dzhepko; YU. G. Nikonorov

paper

Invariant Einstein metrics on some homogeneous spaces of classical Lie groups

arXiv:math/0612504

Abstract

A Riemannian manifold $(M, ρ)$ is called Einstein if the metric $ρ$ satisfies the condition $\Ric (ρ)=c\cdot ρ$ for some constant $c$ . This paper is devoted to the investigation of $G$ -invariant Einstein metrics with additional symmetries, on some homogeneous spaces $G / H$ of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds $S O (n) / S O (l)$ , and on the symplectic analogues $S p (n) / S p (l)$ . Furthermore, we show that for any positive integer $p$ there exists a Stiefel manifold $S O (n) / S O (l)$ and a homogenous space $S p (n) / S p (l)$ which admit at least $p$ SO(n) (resp. $S p (n)$ )-invariant Einstein metrics.