Paper · Feb 1, 2022

Bismut Ricci flat manifolds with symmetries

Fabio Podestà, Alberto Raffero

math.DGarXiv:2202.00417
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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 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.

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Submitted
Feb 1, 2022
Subjects
math.DG
Journal
Proceedings of the Royal Society of Edinburgh. Section A: Mathematics, Vol. 153, Issue 4 (2023) pp. 1371 - 1390
DOI
10.1017/prm.2022.49

Comments: 16 pages

Ricci flowGeneralized & G₂ flowsRicci-flat & Calabi–YauHomogeneous spacesStability & uniqueness

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  1. Generalized Ricci flow on aligned homogeneous spaces

    Valeria Gutiérrez

    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Homogeneous spaces · Shared terms “Bismut Ricci”, “fixed point”

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

    Jorge Lauret and Cynthia E. Will

    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Homogeneous spaces · Shared terms “Bismut Ricci”, “compact homogeneous”, “fixed point”

  3. Three-dimensional positively curved generalized Ricci solitons with SO(3)-symmetries

    Fabio Podestà and Alberto Raffero

    Topics Generalized & G₂ flows, Homogeneous spaces · Same authors Fabio Podestà, Alberto Raffero

  4. Homogeneous Generalized Ricci flows II

    Elia Fusi, Ramiro A. Lafuente and James Stanfield

    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Homogeneous spaces · Shared terms “Bismut”, “construct examples”

  5. Generalized scalar curvature and Ricci flow on compact Lie groups

    Zhiqi Chen, Jianhua Yu, Huibin Chen et al.

    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Homogeneous spaces · Shared terms “Bismut Ricci”, “invariant Bismut”

  6. Some remarks on strong -structures with torsion

    Anna Fino and Udhav Fowdar

    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows · Shared terms “corresponds”

  7. Cohomogeneity one solutions of the IIB system

    Lorenzo Foscolo and Mario Garcia-Fernandez

    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Homogeneous spaces · Shared terms “Bismut”

  8. The Riemannian curvature identities of a connection with skew-symmetric torsion and generalized Ricci solitons

    Stefan Ivanov and Nikola Stanchev

    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 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.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.