Paper · Jul 27, 2020

Finslerian geodesics on Fréchet manifolds

Kaveh Eftekharinasab, Valentyna Petrusenko

math.DGarXiv:2007.13832
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Abstract

We establish a framework, namely, nuclear bounded Fréchet manifolds endowed with Riemann-Finsler structures to study geodesic curves on certain infinite dimensional manifolds such as the manifold of Riemannian metrics on a closed manifold. We prove on these manifolds geodesics exist locally and they are length minimizing in a sense. Moreover, we show that a curve on these manifolds is geodesic if and only if it satisfies a collection of Euler-Lagrange equations. As an application, without much difficulty, we prove that the solution to the Ricci flow on an Einstein manifold is not geodesic.

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Submitted
Jul 27, 2020
Subjects
math.DG
Journal
Bulletin of the Transilvania University of Brasov, Series III: Mathematics, Informatics, Physics, Vol 13(62), No.1 (2020),29-152
DOI
10.31926/but.mif.2020.13.62.1.11
Ricci flowEinstein metrics

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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 A simple proof of the short-time existence and uniqueness for Ricci flow (Kaveh Eftekharinasab, 2016)1Eftekharinas 2016 ε-regularity for shrinking Ricci solitons and Ricci flows (Huabin Ge and Wenshuai Jiang, 2017)4Ge 2017 Complete Ricci solitons on Finsler manifolds (Behroz Bidabad and Mohammad Yar Ahmadi, 2018)2Bidabad 2018 On the long-time behavior of immortal Ricci flows (Shaosai Huang, 2019)3Huang 2019 Li-Yau Estimates for a Nonlinear Parabolic Equation on Finsler Manifolds (Bin Shen and Yuhan Zhu, 2023)6Shen 2023 The continuity of curvature flow in Fréchet distance (Shiyi Ma, 2023)7Ma 2023 Ricci flow and the scalar curvature rigidity of Einstein manifolds (Klaus Kroencke, 2025)8Kroencke 2025 Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds (Frederick Tsz-Ho Fong and Hung Tran, 2026)5Fong 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A simple proof of the short-time existence and uniqueness for Ricci flow

    Kaveh Eftekharinasab

    Topics Einstein metrics · Same author Kaveh Eftekharinasab · Shared terms “Riemannian metrics”

  2. Complete Ricci solitons on Finsler manifolds

    Behroz Bidabad and Mohammad Yar Ahmadi

    Topics Einstein metrics · Shared terms “Finslerian”

  3. On the long-time behavior of immortal Ricci flows

    Shaosai Huang

    Topics Einstein metrics · Shared terms “closed manifold”

  4. -regularity for shrinking Ricci solitons and Ricci flows

    Huabin Ge and Wenshuai Jiang

    Topics Einstein metrics · Shared terms “dimensional manifolds”

  5. Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds

    Frederick Tsz-Ho Fong and Hung Tran

    Topics Einstein metrics

  6. Li-Yau Estimates for a Nonlinear Parabolic Equation on Finsler Manifolds

    Bin Shen and Yuhan Zhu

    Shared terms “Finslerian”, “Euler Lagrange”, “Lagrange equation”

  7. The continuity of curvature flow in Fréchet distance

    Shiyi MaWider flows

    Shared terms “Fréchet”

  8. Ricci flow and the scalar curvature rigidity of Einstein manifolds

    Klaus Kroencke

    Topics Einstein metrics

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 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.

  2. 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.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.