Invariant connections and [formula omitted]-Einstein structures on isotropy irreducible spaces

Abstract

This paper is devoted to a systematic study and classification of invariant affine or metric connections on certain classes of naturally reductive spaces. For any non-symmetric, effective, strongly isotropy irreducible22We will use the abbreviation SII for strongly isotropy irreducible.homogeneous Riemannian manifold (M=G∕K,g), we compute the dimensions of the spaces of G-invariant affine and metric connections. For such manifolds we also describe the space of invariant metric connections with skew-torsion. For the compact Lie group Un we classify all bi-invariant metric connections, by introducing a new family of bi-invariant connections whose torsion is of vectorial type. Next we present applications related with the notion of ∇-Einstein manifolds with skew-torsion. In particular, we classify all such invariant structures on any non-symmetric strongly isotropy irreducible homogeneous space.