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A perspective on totally geodesic submanifolds of the symmetric space $G_2/SO(4)$

Cristina Draper, Cándido Martín-González

TL;DR

The article provides an independent, structure-preserving classification of maximal totally geodesic submanifolds in the exceptional symmetric spaces G2 and G2/SO(4), emphasizing natural, explicit descriptions via Lie triple systems, principal subalgebras, stabilizers, and associative subalgebras of R^7. It develops two complementary frameworks: an octonion-based approach rooted in the cross product and stabilizers of a 3-form Ω, and an octonion-free Grassmannian perspective that describes tangent Lie triple systems in matrix form (notably relating to 𝔰𝔩3(ℝ)). The main contributions include identifying all maximal Lie triple subsystems of 𝔤2 (principal, h2^ℓ, h4^V, and m4^V), and detailing the corresponding maximal totally geodesic submanifolds of G2 and G2/SO(4) with explicit geometric and homogeneous descriptions. The results align with and refine prior classifications by Klein and Kollross, providing concrete, calculable realizations of the submanifolds and their tangent spaces, and revealing the deep interplay between octonionic algebra, cross products, and Grassmannian geometry in exceptional symmetric spaces.

Abstract

We provide an independent proof of the classification of the maximal totally geodesic submanifolds of the symmetric spaces $G_2$ and $G_2/SO(4)$, jointly with very natural descriptions of all of these submanifolds. The description of the totally geodesic submanifolds of $G_2$ is in terms of (1) principal subalgebras of $\mathfrak{g}_2$; (2) stabilizers of nonzero points of $\mathbb{R}^7$; (3) stabilizers of associative subalgebras; (4) the set of order two elements in $G_2$ (and its translations). The space $G_2/SO(4)$ is identified with the set of associative subalgebras of $\mathbb{R}^7$ and its maximal totally geodesic submanifolds can be described as the associative subalgebras adapted to a fixed principal subalgebra, the associative subalgebras orthogonal to a fixed nonzero vector, the associative subalgebras containing a fixed nonzero vector, and the associative subalgebras intersecting both a fixed associative subalgebra and its orthogonal. A second description is included in terms of Grassmannians, the advantage of which is that the associated Lie triple systems are easily described in matrix form.

A perspective on totally geodesic submanifolds of the symmetric space $G_2/SO(4)$

TL;DR

The article provides an independent, structure-preserving classification of maximal totally geodesic submanifolds in the exceptional symmetric spaces G2 and G2/SO(4), emphasizing natural, explicit descriptions via Lie triple systems, principal subalgebras, stabilizers, and associative subalgebras of R^7. It develops two complementary frameworks: an octonion-based approach rooted in the cross product and stabilizers of a 3-form Ω, and an octonion-free Grassmannian perspective that describes tangent Lie triple systems in matrix form (notably relating to 𝔰𝔩3(ℝ)). The main contributions include identifying all maximal Lie triple subsystems of 𝔤2 (principal, h2^ℓ, h4^V, and m4^V), and detailing the corresponding maximal totally geodesic submanifolds of G2 and G2/SO(4) with explicit geometric and homogeneous descriptions. The results align with and refine prior classifications by Klein and Kollross, providing concrete, calculable realizations of the submanifolds and their tangent spaces, and revealing the deep interplay between octonionic algebra, cross products, and Grassmannian geometry in exceptional symmetric spaces.

Abstract

We provide an independent proof of the classification of the maximal totally geodesic submanifolds of the symmetric spaces and , jointly with very natural descriptions of all of these submanifolds. The description of the totally geodesic submanifolds of is in terms of (1) principal subalgebras of ; (2) stabilizers of nonzero points of ; (3) stabilizers of associative subalgebras; (4) the set of order two elements in (and its translations). The space is identified with the set of associative subalgebras of and its maximal totally geodesic submanifolds can be described as the associative subalgebras adapted to a fixed principal subalgebra, the associative subalgebras orthogonal to a fixed nonzero vector, the associative subalgebras containing a fixed nonzero vector, and the associative subalgebras intersecting both a fixed associative subalgebra and its orthogonal. A second description is included in terms of Grassmannians, the advantage of which is that the associated Lie triple systems are easily described in matrix form.

Paper Structure

This paper contains 21 sections, 19 theorems, 67 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Lie triple systems
  4. Curvature and triple systems
  5. Cross products and $G_2$
  6. The exceptional 8-dimensional symmetric space $G_2/\mathop{\hbox{\rm SO}}(4)$
  7. Maximal totally geodesic submanifolds of $G_2$
  8. Some considerations on principal subalgebras
  9. Lie triple systems of $\mathfrak{g}_2$
  10. Maximal totally geodesic submanifolds of $G_2$
  11. Maximal totally geodesic submanifolds of $G_2/\mathop{\hbox{\rm SO}}(4)$
  12. Lie triple system related to the symmetric space $\mathfrak{M}_S$ and its envelope
  13. Adapted principal subalgebras
  14. Maximal Lie triple subsystems
  15. Maximal totally geodesic submanifolds of $\mathfrak{M}_S$
  16. ...and 6 more sections

Key Result

Lemma 1

Take $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ a reductive decomposition of a real or complex Lie algebra $\mathfrak{g}$ (that is, $\mathfrak{h}$ is subalgebra and $[\mathfrak{h},\mathfrak{m}]\subset\mathfrak{m}$). If $\mathfrak{h}$ is semisimple and $\mathfrak{m}$ is an irreducible $\mathfrak{h

Theorems & Definitions (49)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Remark 3
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 39 more