Bismut Ricci flat manifolds with symmetries

TL;DR

This paper constructs examples of compact homogeneous manifolds with invariant Bismut connections that are Ricci flat but non-flat, challenging existing theorems and classifying such spaces in dimension five.

Contribution

It provides new examples of Bismut Ricci flat manifolds, disproves a generalized theorem, and classifies certain spaces, advancing understanding of Bismut geometry.

Findings

Existence of compact homogeneous Bismut Ricci flat manifolds that are non-flat.

Classification of 5-dimensional compact homogeneous Bismut Ricci flat spaces.

Discussion on stability of these manifolds under the generalized Ricci flow.

Abstract

We construct examples of compact homogeneous Riemannian manifolds admitting an invariant Bismut connection that is Ricci flat and non-flat, proving in this way that the generalized Alekseevsky-Kimelfeld theorem does not hold. The classification of compact homogeneous Bismut Ricci flat spaces in dimension $5$ is also provided. Moreover, we investigate compact homogeneous spaces with non trivial third Betti number, and we point out other possible ways to construct Bismut Ricci flat manifolds. Finally, since Bismut Ricci flat connections correspond to fixed points of the generalized Ricci flow, we discuss the stability of some of our examples under the flow.