Paper · Jan 5, 2024

Riemannian Geometry of -type Real Flag Manifolds

Brian Grajales, Gabriel Rondón, Julieth Saavedra

math.DGarXiv:2401.02805v2
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Abstract

In this paper, we investigate homogeneous Riemannian geometry on real flag manifolds of the split real form of . We characterize the metrics that are invariant under the action of a maximal compact subgroup of Our exploration encompasses the analysis of g.o. metrics and equigeodesics on the -type flag manifolds. Additionally, we explore the Ricci flow for the case where the isotropy representation has no equivalent summands, employing techniques from the qualitative theory of dynamical systems.

Comments: 29 pages, 4 figures

Ricci flowHomogeneous spaces

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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 2021, 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 embeddability of the homogeneous Ricci flow and its collapses (Mauro Patrão, Lucas Seco and Llohann D. Sperança, 2021)4Patrão 2021 Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds (A. Loi and R. Mossa, 2023)5Loi 2023 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 Finite extinction time of a family of homogeneous Ricci flows (Roberto Araujo, 2025)7Araujo 2025 The dynamical Alekseevskii conjecture in dimension five (Maher Billon, 2025)8Billon 2025 Universal embeddings of flag manifolds and rigidity phenomena (Andrea Loi, Roberto Mossa and Fabio Zuddas, 2025)2Loi 2025 Homogeneous spaces with two equivalent isotropy summands develop positive Ricci curvature under Ricci flow (Eric Cochran, Arseny Mingajev, Lawrence Mouillé et al., 2026)1Cochran 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Homogeneous spaces with two equivalent isotropy summands develop positive Ricci curvature under Ricci flow

    Eric Cochran, Arseny Mingajev, Lawrence Mouillé et al. PDF

    Topics Homogeneous spaces · Shared terms “isotropy representation”, “equivalent”

  2. Universal embeddings of flag manifolds and rigidity phenomena

    Andrea Loi, Roberto Mossa and Fabio Zuddas PDF

    Topics Homogeneous spaces · 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 PDF

    Topics Homogeneous spaces · Shared terms “flag manifold”

  4. On the embeddability of the homogeneous Ricci flow and its collapses

    Mauro Patrão, Lucas Seco and Llohann D. Sperança PDF

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

  5. Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds

    A. Loi and R. Mossa PDF

    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. PDF

    Topics Ricci flow · Shared terms “flag manifold”

  7. Finite extinction time of a family of homogeneous Ricci flows

    Roberto Araujo PDF

    Topics Homogeneous spaces · Shared terms “homogeneous Riemannian”, “dynamical”

  8. The dynamical Alekseevskii conjecture in dimension five

    Maher Billon PDF

    Topics Homogeneous spaces · Shared terms “dynamical”

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.