New Examples of Homogeneous Einstein Metrics

A Riemannian metric is said to be Einstein if the Ricci curvature is a constant multiple of the metric. Given a manifold M , one can ask whether M carries an Einstein metric, and if so, how many. This fundamental question in Riemannian geometry is for the most part unsolved (cf. [Bes]). As a global PDE or a variational problem, the question is intractible. It becomes more manageable in the homogeneous setting, and so many of the known examples of compact simply connected Einstein manifolds are homogeneous. In this paper we give a technique for nding and classifying all homogeneous metrics on any given homogeneous space, including those which are not diagonal with respect to the isotropy representation. We also examine some compact simply connected homogeneous spaces G=H, where G is simple and H is closed and connected. On each space we describe all G-invariant Einstein metrics. For such spaces, the normal homogeneous Einstein metrics were classi ed by Wang and Ziller [W-Z1]. Among the metrics we nd below, there is only one normal metric: the metric on S S induced by the Killing form. In fact, apart from S S, none of our examples below of homogeneous Einstein metrics is even naturally reductive. Each of our examples has G-invariant metrics which are not diagonal with respect to the isotropy representation of H. Few examples of this type have been previously examined. Some non-diagonal examples arise as brations with Riemannian submersion metrics, where the base and bre are Einstein, e.g., if the base and bre are irreducible symmetric spaces. Using this method, we can expect a product Einstein metric on each of the examples below. Jensen does this to nd a homogeneous Einstein metric on Stiefel manifolds VkR n . He restricts to a two-parameter family of diagonal SO(n)-invariant metrics on VkR n [Je2]. Using very di erent methods, Sagle also considers Stiefel manifolds, showing that VkR n carries at least one Einstein metric [S]. Sagle rst discovered the SO(n)-invariant Einstein metric on V2R n . Neither Sagle nor Jensen observes that the homogeneous Einstein metric on V2R n is unique. More recently, Arvanitoyeorgos looks at a special family of SO(n)-invariant metrics on VkR n [A]. None of these methods exhausts all possible homogeneous Einstein metrics. Our examples consist of three symmetric spaces and the unit tangent bundle of the n-sphere. We nd the following: