Invariant connections and $\nabla$-Einstein structures on isotropy irreducible spaces

TL;DR

This paper systematically classifies invariant affine and metric connections, including those with skew-torsion, on certain homogeneous spaces, and explores their applications to $ abla$-Einstein structures, revealing new connections on Lie groups.

Contribution

It provides a comprehensive classification of invariant connections on non-symmetric strongly isotropy irreducible spaces and introduces new bi-invariant connections with vectorial torsion.

Findings

Dimensions of invariant affine and metric connection spaces computed

Classification of invariant metric connections with skew-torsion

New bi-invariant connections with vectorial torsion introduced

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 ${\rm U}_{n}$ 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.