Paper · Jan 5, 2023

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

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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 $r i c (g) = \frac{1}{4} H_{g}^{2}$ . 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 $G_{1} / π_{1} (K)$ and $G_{2} / π_{2} (K)$ , 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 $M = G \times G /Δ K$ and for $M^{35} = S O (8) \times S O (7) / G_{2}$ . A Corrigendum has been added in Appendix A.

Comments: 27 pages. Final version published in Transactions AMS with a Corrigendum added

Ricci flow Generalized & G₂ flows Ricci-flat & Calabi–Yau Smale conjecture & diffeomorphisms Homogeneous spaces Eigenvalues & spectrum

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12C.9
After PerelmanBook 12C · Extinction and the Proof
Singular flows, the Smale conjecture, and Bamler’s theory.
11A.13
Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles
A computable laboratory: the flow on Thurston’s geometries.
5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

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

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