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Riemannian Geometry of $G_2$-type Real Flag Manifolds

Brian Grajales, Gabriel Rondón, Julieth Saavedra

Abstract

In this paper, we investigate homogeneous Riemannian geometry on real flag manifolds of the split real form of $\mathfrak{g}_2$. We characterize the metrics that are invariant under the action of a maximal compact subgroup of $G_2.$ Our exploration encompasses the analysis of g.o. metrics and equigeodesics on the $\mathfrak{g}_2$-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.

Riemannian Geometry of $G_2$-type Real Flag Manifolds

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.
Paper Structure (16 sections, 15 theorems, 138 equations, 4 figures, 1 table)

This paper contains 16 sections, 15 theorems, 138 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compact homogeneous spaces
  4. The non-compact Lie algebra $\mathfrak{g}_2$
  5. Real flag manifolds of $\mathfrak{g}_2$
  6. Homogeneous geodesics on flag manifolds
  7. $K$-invariant metrics
  8. G.o. metrics
  9. Equigeodesics
  10. The homogeneous Ricci flow
  11. Local dynamic
  12. Dynamics around $q_1$
  13. Dynamics around $q_2$
  14. Dynamics around $q_3$
  15. Dynamics around $q_4$
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $\mathbb{F}_\Theta$ be a real flag manifold of $\mathfrak{g}_2.$ Then the following statements hold:

Figures (4)

  • Figure 1: Phase portrait of system \ref{['main_eq_1']}.
  • Figure 2: Projection of the phase portrait of system \ref{['main_eq_1']} onto the $yz-$plane.
  • Figure 3: Projection of the phase portrait of system \ref{['main_eq_1']} onto the $xz-$plane.
  • Figure 4: Projection of the phase portrait of system \ref{['main_eq_1']} onto the $xy-$plane.

Theorems & Definitions (30)

  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Theorem 3.1
  • ...and 20 more