Einstein connections with skew-torsion on Berger spheres

TL;DR

This paper classifies invariant metric affine connections with Einstein property and skew torsion on Berger spheres, providing explicit formulas and analyzing existence conditions in both Riemannian and Lorentzian signatures.

Contribution

It explicitly determines and characterizes Einstein affine connections with skew torsion on Berger spheres in both signatures, highlighting dimension and deformation dependencies.

Findings

Lorentzian Berger spheres admit such connections up to b^3

Existence in Riemannian case depends on dimension and deformation scale

Some non-Einstein Riemannian Berger spheres admit Einstein connections with skew torsion

Abstract

The invariant metric affine connections on Berger spheres which are Einstein with skew torsion are determined in both Riemannian and Lorentzian signature. Expressions of such connections are explicitly given. In particular, every Berger sphere with Lorentzian signature admits invariant metric affine connections Einstein with skew-torsion up to $\mathbb{S}^3$. For Riemannian signature, the existence of such connections strongly depends on the dimension of the sphere and on the scale of the deformation used for the Berger metric. In particular, there are Riemaniann Berger spheres, not Einstein, which admit invariant metric affine connections Einstein with skew-torsion.