Paper · Dec 29, 2019

Diagonalizing the Ricci Tensor

Anusha M. Krishnan

math.DGarXiv:1912.12686v2
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Abstract

We show that a basis of a semisimple Lie algebra of compact type, for which any diagonal left-invariant metric has a diagonal Ricci tensor, is characterized by the Lie algebraic condition of being "nice". Namely, the bracket of any two basis elements is a multiple of another basis element. This extends the work of Lauret and Will on nilpotent Lie algebras. The result follows from a more general characterization for diagonalizing the Ricci tensor for homogeneous spaces. Finally, we also study the Ricci flow behavior of diagonal metrics on cohomogeneity one manifolds.

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Submitted
Dec 29, 2019
Last revised
Dec 29, 2021 (v2)
Subjects
math.DG
Journal
J. Geom. Anal. 31 (2021), no. 6, 5638-5658

Comments: LaTeX2e, 17 pages, final version

Ricci flowHomogeneous spaces

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Laplacian flow of homogeneous G2-structures and its solitons (Jorge Lauret, 2016)8Lauret 2016 Four-dimensional cohomogeneity one Ricci flow and nonnegative sectional curvature (Renato G. Bettiol and Anusha M. Krishnan, 2016)3Bettiol 2016 Ricci flow does not preserve positive sectional curvature in dimension four (Renato G. Bettiol and Anusha M. Krishnan, 2021)7Bettiol 2021 A Hilbert–Mumford criterion for nilsolitons (Yoshinori Hashimoto, 2023)6Hashimoto 2023 Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow (Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti, 2023)1Krishnan 2023 Nilpotent Lie algebras obtained by quivers and Ricci solitons (Fumika Mizoguchi and Hiroshi Tamaru, 2024)5Mizoguchi 2024 On the long time behavior of ancient homogeneous Ricci flows (Anusha M. Krishnan and Francesco Pediconi, 2026)2Krishnan 2026 Nilpotent Lie algebras obtained by ordered sets and Ricci solitons (Yihao Zheng and Shenglin Zhu, 2026)4Zheng 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow

    Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti

    Topics Homogeneous spaces · Same author Anusha M. Krishnan · Shared terms “diagonal metrics”, “algebraic”

  2. On the long time behavior of ancient homogeneous Ricci flows

    Anusha M. Krishnan and Francesco Pediconi

    Topics Homogeneous spaces · Same author Anusha M. Krishnan

  3. Four-dimensional cohomogeneity one Ricci flow and nonnegative sectional curvature

    Renato G. Bettiol and Anusha M. Krishnan

    Topics Homogeneous spaces · Same author Anusha M. Krishnan · Shared terms “cohomogeneity”

  4. Nilpotent Lie algebras obtained by ordered sets and Ricci solitons

    Yihao Zheng and Shenglin Zhu

    Topics Homogeneous spaces · Shared terms “Lie algebras”, “nilpotent Lie”, “left invariant”

  5. Nilpotent Lie algebras obtained by quivers and Ricci solitons

    Fumika Mizoguchi and Hiroshi Tamaru

    Topics Homogeneous spaces · Shared terms “Lie algebras”, “nilpotent Lie”, “left invariant”

  6. A Hilbert–Mumford criterion for nilsolitons

    Yoshinori Hashimoto

    Topics Homogeneous spaces · Shared terms “Lie algebras”, “nilpotent Lie”, “left invariant”

  7. Ricci flow does not preserve positive sectional curvature in dimension four

    Renato G. Bettiol and Anusha M. Krishnan

    Topics Homogeneous spaces · Same author Anusha M. Krishnan · Shared terms “cohomogeneity”

  8. Laplacian flow of homogeneous G2-structures and its solitons

    Jorge Lauret

    Topics Homogeneous spaces · Shared terms “diagonal”, “algebraic”, “nilpotent Lie”

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) 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.