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Generalized Ricci flow on aligned homogeneous spaces

Valeria Gutiérrez

TL;DR

This work develops and analyzes the generalized Ricci flow (GRF) on aligned homogeneous spaces $M=G_1\times G_2/\Delta K$, where each $G_i$ is a compact simple group and $K$ is embedded diagonally. It confirms the existence of Bismut Ricci flat (BRF) invariant metrics $(g_0,H_0)$ on these spaces and proves dynamical stability of these fixed points: locally for diagonal $G$-invariant metrics and globally when $G_1=G_2$. The analysis relies on reducing GRF to finite-dimensional ODEs in diagonal metric coordinates, leveraging the BRF structure and the Einstein property of the factor spaces, and deploying linearization and Lyapunov techniques to establish asymptotic and global stability. In the SO$(n)$ case, the flow near the Killing metric is shown to converge to the BRF structure, supporting the conjecture that BRF metrics are dynamically attractive under GRF in a broad class of compact Lie groups.

Abstract

The fixed points of the generalized Ricci flow are the Bismut Ricci flat metrics, i.e., a generalized metric $(g,H)$ on a manifold $M$, where $g$ is a Riemannian metric and $H$ a closed $3$-form, such that $H$ is $g$-harmonic and $\operatorname{Rc}(g)=\tfrac{1}{4} H_g^2$. Given two standard Einstein homogeneous spaces $G_i/K$, where each $G_i$ is a compact simple Lie group and $K$ is a closed subgroup of them holding some extra assumption, we consider $M = G_1 \times G_2 / ΔK$. Recently, Lauret and Will proved the existence of a Bismut Ricci flat metric on any of these spaces. We proved that this metric is always asymptotically stable for the generalized Ricci flow on $M$ among a subset of $G$-invariant metrics and, if $G_1 = G_2$, then it is globally stable.

Generalized Ricci flow on aligned homogeneous spaces

TL;DR

This work develops and analyzes the generalized Ricci flow (GRF) on aligned homogeneous spaces , where each is a compact simple group and is embedded diagonally. It confirms the existence of Bismut Ricci flat (BRF) invariant metrics on these spaces and proves dynamical stability of these fixed points: locally for diagonal -invariant metrics and globally when . The analysis relies on reducing GRF to finite-dimensional ODEs in diagonal metric coordinates, leveraging the BRF structure and the Einstein property of the factor spaces, and deploying linearization and Lyapunov techniques to establish asymptotic and global stability. In the SO case, the flow near the Killing metric is shown to converge to the BRF structure, supporting the conjecture that BRF metrics are dynamically attractive under GRF in a broad class of compact Lie groups.

Abstract

The fixed points of the generalized Ricci flow are the Bismut Ricci flat metrics, i.e., a generalized metric on a manifold , where is a Riemannian metric and a closed -form, such that is -harmonic and . Given two standard Einstein homogeneous spaces , where each is a compact simple Lie group and is a closed subgroup of them holding some extra assumption, we consider . Recently, Lauret and Will proved the existence of a Bismut Ricci flat metric on any of these spaces. We proved that this metric is always asymptotically stable for the generalized Ricci flow on among a subset of -invariant metrics and, if , then it is globally stable.
Paper Structure (10 sections, 13 theorems, 56 equations, 5 figures)

This paper contains 10 sections, 13 theorems, 56 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Aligned homogeneous spaces
  4. Bismut connection and generalized Ricci flow
  5. Bismut Ricci flat metrics on aligned homogeneous spaces
  6. Generalized Ricci flow on aligned spaces
  7. Global stability
  8. Generalized Ricci flow on the Lie group $\mathrm{SO}(n)$
  9. Set up for any compact Lie group
  10. Case $\mathrm{SO}(n)$

Key Result

Theorem 1.1

Let $M = G_1 \times G_2/\Delta K$ be a homogeneous space as above such that $\operatorname{B}_{\mathfrak{k} }=a_i \operatorname{B}_{\mathfrak{g} _i}|_{\mathfrak{k} }$ and $(M_i=G_i/K, g_{\operatorname{B}}^i)$ is Einstein, where $g_{\operatorname{B}}^i$ is the standard metric on each $M_i$ for $i=1,2

Figures (5)

  • Figure 1: Flow in the invariant plane $x_3=\tfrac{10}{3}$.
  • Figure 2: Flow in the invariant plane $x_1=\tfrac{3}{7}x_2$.
  • Figure 3: Region of Case 1: $g_1<0$ and $g_2<0$.
  • Figure 4: Region of Case 2: $g_1<0$ and $g_2>0$.
  • Figure 5: Region of Case 3: $g_1>0$ and $g_2<0$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Lemma 3.1
  • proof
  • ...and 14 more