Einstein homogeneous bisymmetric fibrations

Abstract

We consider a homogeneous fibration G/L → G/K, with symmetric fiber and base, where G is a compact connected semisimple Lie group and L has maximal rank in G. We suppose the base space G/K is isotropy irreducible and the fiber K/L is simply connected. We investigate the existence of G-invariant Einstein metrics on G/L such that the natural projection onto G/K is a Riemannian submersion with totally geodesic fibers. These spaces are divided in two types: the fiber K /L is isotropy irreducible or is the product of two irreducible symmetric spaces. We classify all the G-invariant Einstein metrics with totally geodesic fibers for the first type. For the second type, we classify all these metrics when G is an exceptional Lie group. If G is a classical Lie group we classify all such metrics which are the orthogonal sum of the normal metrics on the fiber and on the base or such that the restriction to the fiber is also Einstein.

Non-naturally reductive Einstein metrics on normal homogeneous Einstein manifolds

“…The method of Riemannian submersion is an important tool to construct new examples of Einstein metrics, and it has been applied to obtain many interesting existence results; see Chapter 9 of [4] and some results in [1,2,10]. Let G/H be a compact connected homogeneous space, and g = h + m a reductive decomposition of g, where g, h denote the Lie algebras of G and H respectively, and m is a subspace of g such that Ad(H)(m) ⊂ m. Then there is a one-to-one correspondence between the G-invariant Riemannian metric on G/H and the Ad(H)-invariant inner product on m. Recall that an invariant metric on G/H is called normal if the corresponding inner product on m is the restriction of a bi-invariant inner product on g. In particular, let B denote the negative Killing form of g, and g B be the standard metric on G/H induced by B| m .…”