Invariant connections with skew-torsion and $\nabla$-Einstein manifolds

Abstract

For a compact connected Lie group $G$ we study the class of bi-invariant affine connections whose geodesics through $e\in G$ are the 1-parameter subgroups. We show that the bi-invariant affine connections which induce derivations on the corresponding Lie algebra $\mathfrak{g}$ coincide with the bi-invariant metric connections. Next we describe the geometry of a naturally reductive space $(M=G/K,g)$ endowed with a family of $G$-invariant connections ${\nabla }^{\alpha }$ whose torsion is a multiple of the torsion of the canonical connection ${\nabla }^c$. For the spheres ${\mathrm{S}}^6$ and ${\mathrm{S}}^7$ we prove that the space of ${\mathrm{G}}_2$ (resp. $\mathrm{Spin}(7)$)-invariant affine or metric connections consists of the family ${\nabla }^{\alpha }$. Then we examine the "constancy" of the induced Ricci tensor ${\mathrm{Ric}}^{\alpha }$ and prove that any compact simply-connected isotropy irreducible standard homogeneous Riemannian manifold, which is not a symmetric space of Type I, is a ${\nabla }^{\alpha }$-Einstein manifold for any $\alpha \in \mathbb{R}$. We also provide examples of ${\nabla }^{\pm 1}$-Einstein structures for a class of compact homogeneous spaces $M=G/K$ with two isotropy summands.

25 pages, to appear in Journal of Lie Theory. The presentation of the paper has been improved, some misprints and errors were corrected. The material regarding the flat case has been removed and an error in Theorem 4.7 has been corrected

Cited by in corpus (1)

• Killing and twistor spinors with torsion