Invariant connections and $\nabla$-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 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 ${\mathrm{U}}_n$ 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 $\nabla$-Einstein manifolds with skew-torsion. In particular, we classify all such invariant structures on any non-symmetric strongly isotropy irreducible homogeneous space.

References in corpus (2)

• Spin structures on compact homogeneous pseudo-Riemannian manifolds
• Killing and twistor spinors with torsion

Cited by in corpus (2)

• Differential geometry of ${\mathsf{SO}}^{\ast }(2n)$-type structures
• A note on the volume of $\nabla$-Einstein manifolds with skew-torsion