Bismut Ricci flat generalized metrics on compact homogeneous spaces (including a Corrigendum)

TL;DR

This paper investigates Bismut Ricci flat generalized metrics on compact homogeneous spaces, establishing existence, uniqueness, and non-homothety of such metrics under various conditions, with explicit examples and a corrigendum.

Contribution

It provides new existence and uniqueness results for Bismut Ricci flat generalized metrics on specific homogeneous spaces, including explicit families and non-homothety properties.

Findings

Existence of unique G-invariant Bismut Ricci flat metrics on certain homogeneous spaces.

Construction of one-parameter families of such metrics with non-homothety properties.

Explicit examples including spaces like G×G/ΔK and SO(8)×SO(7)/G_2.

Abstract

A generalized metric on a manifold $M$, i.e., a pair $(g,H)$, where $g$ is a Riemannian metric and $H$ a closed $3$-form, is a fixed point of the generalized Ricci flow if and only if $(g,H)$ is Bismut Ricci flat: $H$ is $g$-harmonic and $ric(g)=\tfrac{1}{4} H_g^2$. On any homogeneous space $M=G/K$, where $G=G_1\times G_2$ is a compact semisimple Lie group with two simple factors, under some mild assumptions, we exhibit a Bismut Ricci flat $G$-invariant generalized metric, which is proved to be unique among a $4$-parameter space of metrics in many cases, including when $K$ is neither abelian nor semisimple. On the other hand, if $K$ is simple and the standard metric is Einstein on both $G_1/\pi_1(K)$ and $G_2/\pi_2(K)$, we give a one-parameter family of Bismut Ricci flat $G$-invariant generalized metrics on $G/K$ and show that it is most likely pairwise non-homothetic by computing the ratio of Ricci eigenvalues. This is proved to be the case for every space of the form $M=G\times G/\Delta K$ and for $M^{35}=SO(8)\times SO(7)/G_2$. A Corrigendum has been added in Appendix A.