Non-naturally reductive Einstein metrics on normal homogeneous Einstein manifolds

A quadruple of Lie groups [Formula: see text], where [Formula: see text] is a compact semisimple Lie group, [Formula: see text] are closed subgroups of [Formula: see text], and the related Casimir constants satisfy certain appropriate conditions, is called a basic quadruple. A basic quadruple is called Einstein if the Killing form metrics on the coset spaces [Formula: see text], [Formula: see text] and [Formula: see text] are all Einstein. In this paper, we first give a complete classification of the Einstein basic quadruples. We then show that, except for very few exceptions, given any quadruple [Formula: see text] in our list, we can produce new non-naturally reductive Einstein metrics on the coset space [Formula: see text], by scaling the Killing form metrics along the complement of [Formula: see text] in [Formula: see text] and along the complement of [Formula: see text] in [Formula: see text]. We also show that on some compact semisimple Lie groups, there exist a large number of left invariant non-naturally reductive Einstein metrics which are not product metrics. This discloses a new interesting phenomenon which has not been described in the literature.

Naturally reductive homogeneous (α,β) spaces

In this paper, we study naturally reductive [Formula: see text]-metrics on homogeneous manifolds. We show that naturally reductive [Formula: see text]-metrics arise only when [Formula: see text] is naturally reductive and some conditions on [Formula: see text] is satisfied. We give an explicit formula for the flag curvature of naturally reductive [Formula: see text]metrics which improves the flag curvature formula of naturally reductive Randers metrics given in [D. Latifi, Naturally reductive homogeneous Randers spaces, J. Geom. Phys. 60 (2010) 1968–1973]. As a special case, we give an explicit formula for the flag curvature of bi-invariant [Formula: see text]-metrics on Lie groups.