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On cohomogeneity one hyperpolar actions related to $G_{2}$

Shinji Ohno, Yuuki Sasaki

TL;DR

The paper investigates exceptional cohomogeneity-one hyperpolar actions arising from $G_{2}$ on compact symmetric spaces, focusing on principal-orbit geometry to identify minimal, austere, weakly reflective, and proper biharmonic submanifolds. Using octonion and Spin(7) formalisms, it computes principal curvatures for principal orbits under types (II)–(V) and determines the special orbits meeting the given curvature criteria. The main results enumerate, for each type, the existence and nature of minimal orbits (often weakly reflective) and count the proper biharmonic principal orbits, with explicit parameter values and curvature decompositions; level-set characterizations of certain $K$-orbits on $G/H$ are also described. Overall, the work advances the understanding of exceptional cohomogeneity-one hyperpolar actions by providing a detailed classification of minimal, austere, weakly reflective, and biharmonic principal orbits in the $G_{2}$-related setting, contributing to the theory of minimal submanifolds and biharmonic submanifolds in compact symmetric spaces.

Abstract

Cohomogeneity one actions on irreducible Riemannian symmetric spaces of compact type are classified into three cases: Hermann actions, actions induced by the linear isotropy representation of a Riemannian symmetric space of rank 2, and exceptional actions. In this paper, we consider exceptional actions related to the exceptional compact Lie group $G_{2}$ and investigate some properties of their orbits as Riemannian submanifolds. In particular, we examine the principal curvatures of principal orbits and classify principal orbits that are minimal, austere, weakly reflective, and proper biharmonic.

On cohomogeneity one hyperpolar actions related to $G_{2}$

TL;DR

The paper investigates exceptional cohomogeneity-one hyperpolar actions arising from on compact symmetric spaces, focusing on principal-orbit geometry to identify minimal, austere, weakly reflective, and proper biharmonic submanifolds. Using octonion and Spin(7) formalisms, it computes principal curvatures for principal orbits under types (II)–(V) and determines the special orbits meeting the given curvature criteria. The main results enumerate, for each type, the existence and nature of minimal orbits (often weakly reflective) and count the proper biharmonic principal orbits, with explicit parameter values and curvature decompositions; level-set characterizations of certain -orbits on are also described. Overall, the work advances the understanding of exceptional cohomogeneity-one hyperpolar actions by providing a detailed classification of minimal, austere, weakly reflective, and biharmonic principal orbits in the -related setting, contributing to the theory of minimal submanifolds and biharmonic submanifolds in compact symmetric spaces.

Abstract

Cohomogeneity one actions on irreducible Riemannian symmetric spaces of compact type are classified into three cases: Hermann actions, actions induced by the linear isotropy representation of a Riemannian symmetric space of rank 2, and exceptional actions. In this paper, we consider exceptional actions related to the exceptional compact Lie group and investigate some properties of their orbits as Riemannian submanifolds. In particular, we examine the principal curvatures of principal orbits and classify principal orbits that are minimal, austere, weakly reflective, and proper biharmonic.

Paper Structure

This paper contains 9 sections, 22 theorems, 82 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some submanifolds
  4. The compact Lie group $G_{2}$
  5. The compact Lie group $Spin(7)$
  6. Type (II)
  7. Type (III)
  8. Type (IV)
  9. Type (V)

Key Result

Lemma 2.3

IST2 Let $G$ be a connected Lie group acting isometrically on $(N, h)$. Denote the $G$-orbit through $x \in N$ by $G(x)$. If for each $\xi \in T_{x}^{\perp}G(x)$, there exists an isometry $\sigma_{\xi}$ of $N$ such that $\sigma_{\xi}(x) = x, d\sigma_{\xi}(\xi) = -\xi,$ and $\sigma_{\xi}(G(x)) = G(x)

Figures (1)

  • Figure 1: Octonions

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Corollary 2.8
  • Lemma 3.1
  • Lemma 3.2
  • ...and 17 more