Paper · Feb 24, 2016

Variational aspects of homogeneous geodesics on generalized flag manifolds and applications

Rafaela F. do Prado, Lino Grama

math.DGarXiv:1602.07660
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Abstract

We study conjugate points along homogeneous geodesics in generalized flag manifolds. This is done by analyzing the second variation of the energy of such geodesics. We also give an example of how the homogeneous Ricci flow can evolve in such way to produce conjugate points in the complex projective space .

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Submitted
Feb 24, 2016
Subjects
math.DG
Ricci flowHomogeneous spaces

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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 On the stability of harmonic maps under the homogeneous Ricci flow (Rafaela F. do Prado and Lino Grama, 2017)1Prado 2017 Ricci flow on certain homogeneous spaces (Marina Statha, 2020)4Statha 2020 The projected homogeneous Ricci flow and three-isotropy-summands flag manifolds (Lino Grama, Ricardo M. Martins, Mauro Patrão et al., 2020)2Grama 2020 Ancient solutions of the homogeneous Ricci flow on flag manifolds (Stavros Anastassiou and Ioannis Chrysikos, 2020)7Anastassiou 2020 On the dynamics of positively curved metrics on SU(3)/T² under the homogeneous Ricci flow (Leonardo F. Cavenaghi, Lino Grama and Ricardo M. Martins, 2023)3Cavenaghi 2023 The complete dynamics description of positively curved metrics in the Wallach flag manifold SU(3)/T² (Leonardo F. Cavenaghi, Lino Grama, Ricardo M. Martins et al., 2023)6Cavenaghi 2023 Riemannian Geometry of G₂-type Real Flag Manifolds (Brian Grajales, Gabriel Rondón and Julieth Saavedra, 2024)5Grajales 2024 Universal embeddings of flag manifolds and rigidity phenomena (Andrea Loi, Roberto Mossa and Fabio Zuddas, 2025)8Loi 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the stability of harmonic maps under the homogeneous Ricci flow

    Rafaela F. do Prado and Lino Grama

    Topics Homogeneous spaces · Same authors Rafaela F. do Prado, Lino Grama

  2. The projected homogeneous Ricci flow and three-isotropy-summands flag manifolds

    Lino Grama, Ricardo M. Martins, Mauro Patrão et al.

    Topics Homogeneous spaces · Same author Lino Grama · Shared terms “flag manifold”

  3. On the dynamics of positively curved metrics on under the homogeneous Ricci flow

    Leonardo F. Cavenaghi, Lino Grama and Ricardo M. Martins

    Topics Homogeneous spaces · Same author Lino Grama · Shared terms “flag manifold”

  4. Ricci flow on certain homogeneous spaces

    Marina Statha

    Topics Homogeneous spaces · Shared terms “generalized flag”, “flag manifold”

  5. Riemannian Geometry of -type Real Flag Manifolds

    Brian Grajales, Gabriel Rondón and Julieth Saavedra

    Topics Homogeneous spaces · Shared terms “flag manifold”

  6. The complete dynamics description of positively curved metrics in the Wallach flag manifold

    Leonardo F. Cavenaghi, Lino Grama, Ricardo M. Martins et al.

    Topics Ricci flow · Same author Lino Grama · Shared terms “flag manifold”

  7. Ancient solutions of the homogeneous Ricci flow on flag manifolds

    Stavros Anastassiou and Ioannis Chrysikos

    Topics Homogeneous spaces · Shared terms “flag manifold”

  8. Universal embeddings of flag manifolds and rigidity phenomena

    Andrea Loi, Roberto Mossa and Fabio Zuddas

    Topics Homogeneous spaces · Shared terms “flag manifold”

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.