Some Homogeneous Einstein Manifolds

Abstract

1. Introduction. Let G be a connected Lie group and H a closed subgroup with Lie algebra ϊ) such that in the Lie algebra g of G there exists a subspace m with g = m + § (subspace direct sum) and [mϊflcnt. In this case the corresponding manifold M = G/H is called a reductive homogeneous space and (g,ϊj) (or {G,H)) a reductive pair. In this paper we shall show how to construct invariant pseudo-Riemannian connections on suitable reductive homogeneous spaces M which make M into an Einstein manifold. Thus from the formula for the connection at the point H^M we compute the Ricci tensor, Ric (X,Y) for X, Fem,Y) = rjC{X,Y) where C(X,Y) is the pseudo-Riemannian metric inducing the connection and η is a suitable real number.

On Einstein Riemannian g-natural metrics on unit tangent sphere bundles

“…Before starting, and as a motivation, let us review the known results about SO(m + 1)-invariant Einstein metrics on the Stiefel manifold V 2 R m+1 . Sagle proved that there is an SO(m + 1)-invariant Einstein metric on V 2 R m+1 = SO(m + 1)/SO(m − 1), except for m = 3 or 7 [23]. But, Sagle was not the first to prove the existence of a such metric on V 2 R m+1 , since this metric is actually a special case of a theorem by Kobayashi which is related with the recently developed rich theory of Sasakian Einstein manifolds.…”

Stability of Riemannian manifolds with Killing spinors

“…The Stiefel manifold V 2 (R n+1 ) = SO(n+1) SO(n−1) (n ≥ 3) may be viewed as a principal circle bundle over the real oriented Grassmannian SO(n+1) SO(n−1)SO(2) , which is an irreducible Hermitian symmetric space (with second Betti number b 2 = 1). Sagle [Sa70] constructed an invariant Einstein metric on V 2 (R n+1 ) which is now known to be unique (up to isometry and homothety) among all SO(n + 1)-invariant metrics [Ker98], except when n = 3. Its relevance for us is that it can be viewed as the regular Sasaki Einstein metric determined by the base considered as a Fano Einstein manifold with the symmetric metric scaled so that its scalar curvature equals 2n + 2.…”

Non-trivial m-quasi-Einstein metrics on quadratic Lie groups

“…There is a simple formula of the Ricci curvature on a compact semisimple Lie group with respect to a left-invariant Riemannian metric which is derived in [19], and a simpler proof is given in [9]. Motivated by the proof in [9], we have a formula for a quadratic Lie group.…”