Bismut Ricci flat manifolds with symmetries

TL;DR

The paper addresses whether invariant BRF pairs on compact homogeneous spaces must yield flat Bismut connections, and constructs non-flat BRF examples on spaces ${\mathrm M}_{p,q}$ (diffeomorphic to $S^3\times S^2$) to disprove a generalized Alekseevsky-Kimelfeld expectation. It combines explicit Lie-algebraic analysis on ${\mathrm M}_{p,q}$, harmonic 3-forms $H$, and Kobayashi-type $S^1$-bundle constructions to produce BRF pairs with ${\rm Ric}_g=\frac14 H^2$ and non-flat $\nabla$, further classifying 5D invariant BRF spaces. Additionally, the generalized Ricci flow (GRF) stability is established: for $p\neq q$ the invariant BRF fixed point is globally attracting, with $H_t$ proportional to $H_0$ and the metric evolving through a closed autonomous system, while the $M_{1,1}$ case requires separate analysis due to off-diagonal terms. The results provide novel BRF manifolds beyond the AK framework and illuminate GRF dynamics on compact homogeneous spaces, with implications for generalized Einstein geometry and potential construction methods via Kobayashi’s approach.

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.

Theorems & Definitions (16)

• Theorem 2.1
• proof
• Remark 2.2
• Proposition 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Remark 3.4