Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Polar actions on homogeneous 3-spaces

Miguel Dominguez-Vazquez, Tarcios A. Ferreira, Tomas Otero

TL;DR

This work classifies polar isometric actions on simply connected 3-dimensional homogeneous spaces up to orbit equivalence, focusing on cohomogeneity-one and cohomogeneity-two actions and the geometry of the resulting homogeneous surfaces. It leverages a complete 3D classification of metric Lie groups (unimodular and non-unimodular) and a polarity criterion to reduce the problem to codimension-one subalgebra data, yielding explicit moduli spaces for cohomogeneity-one actions and a finite list for cohomogeneity-two actions, with detailed extrinsic geometry of the corresponding orbits. The paper provides a thorough analysis of the shape operators and mean curvature of the homogeneous hypersurfaces arising from each action, revealing when orbits are minimal or totally geodesic. Overall, it extends known polar-action classifications from space forms and E(kappa, tau) spaces to all simply connected 3D homogeneous spaces, offering a framework for constructing and studying isoparametric foliations in low dimensions.

Abstract

We classify polar isometric actions on simply connected 3-dimensional Riemannian homogeneous spaces, up to orbit equivalence. In particular, we classify extrinsically homogeneous surfaces in such spaces and study the geometry of the orbit foliations of the corresponding cohomogeneity one actions.

Polar actions on homogeneous 3-spaces

TL;DR

This work classifies polar isometric actions on simply connected 3-dimensional homogeneous spaces up to orbit equivalence, focusing on cohomogeneity-one and cohomogeneity-two actions and the geometry of the resulting homogeneous surfaces. It leverages a complete 3D classification of metric Lie groups (unimodular and non-unimodular) and a polarity criterion to reduce the problem to codimension-one subalgebra data, yielding explicit moduli spaces for cohomogeneity-one actions and a finite list for cohomogeneity-two actions, with detailed extrinsic geometry of the corresponding orbits. The paper provides a thorough analysis of the shape operators and mean curvature of the homogeneous hypersurfaces arising from each action, revealing when orbits are minimal or totally geodesic. Overall, it extends known polar-action classifications from space forms and E(kappa, tau) spaces to all simply connected 3D homogeneous spaces, offering a framework for constructing and studying isoparametric foliations in low dimensions.

Abstract

We classify polar isometric actions on simply connected 3-dimensional Riemannian homogeneous spaces, up to orbit equivalence. In particular, we classify extrinsically homogeneous surfaces in such spaces and study the geometry of the orbit foliations of the corresponding cohomogeneity one actions.
Paper Structure (21 sections, 4 theorems, 51 equations, 6 tables)

This paper contains 21 sections, 4 theorems, 51 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Riemannian homogeneous $3$-manifolds and metric Lie groups
  3. Unimodular Lie groups of dimension $3$
  4. Non-unimodular Lie groups of dimension $3$
  5. Cohomogeneity one actions on $3$-dimensional groups
  6. Codimension one subalgebras of unimodular Lie algebras
  7. Codimension one subalgebras of $\mathfrak{e}_2$.
  8. Codimension one subalgebras of $\mathfrak{sl}_2(\mathbb{R})$.
  9. Codimension one subalgebras of $\mathfrak{sol}_3$.
  10. Codimension one subalgebras of non-unimodular Lie algebras
  11. Cohomogeneity two polar actions
  12. Polar actions on metric Lie groups
  13. Polar actions on symmetric spaces
  14. Geometry of the $2$-dimensional homogeneous foliations
  15. Homogeneous surfaces of $\widetilde{\mathsf{E}}_2$.
  16. ...and 6 more sections

Key Result

Theorem A

An isometric action of a connected Lie group on a $3$-dimensional simply connected metric Lie group $G$ with $\dim\left(\mathop{\mathrm{Isom}}\nolimits(G)\right)=3$ is of cohomogeneity one if and only if it is orbit equivalent to the action of the connected Lie subgroup $H$ of $G$ with one (and only

Theorems & Definitions (11)

  • Theorem A
  • Theorem B
  • Remark 2.1
  • Remark 2.2
  • Remark 3.1: Cohomogeneity one actions on space forms
  • Remark 3.2: Cohomogeneity one actions on $\mathbb{E}(\kappa,\tau)$-spaces
  • Proposition 3.3
  • Remark 3.4
  • Remark 4.1
  • Lemma 4.2
  • ...and 1 more