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Homogeneous hypersurfaces of the four-dimensional Thurston geometry ${\rm Sol_0^4}$

Marie D'haene, Guoxin Wei, Zeke Yao, Xi Zhang

TL;DR

This work determines all homogeneous hypersurfaces in the four-dimensional Thurston geometry $Sol_0^4$ by classifying hypersurfaces with constant principal curvatures under constant angle functions. The authors develop the ambient $Sol_0^4$ geometry, derive the Gauss-Codazzi equations for hypersurfaces in this space, and identify four explicit model families $M_{1,r},M_{2,r},M_{3,r},M_{4,0}$ with precise principal curvatures. They then prove that any homogeneous hypersurface in $Sol_0^4$ is congruent to one of these models, thus solving a standing problem in EI2. The results provide explicit parametrizations, curvature data, and orbit descriptions, advancing the understanding of isoparametric and homogeneous submanifolds in non-symmetric Thurston geometries.

Abstract

In this paper, we classify hypersurfaces with constant principal curvatures in the four-dimensional Thurston geometry ${\rm Sol_0^4}$ under certain geometric conditions. As an application of the classification result, we give a complete classification of homogeneous hypersurfaces in ${\rm Sol_0^4}$, which solves a problem raised by Erjavec and Inoguchi (Problem 6.4 of [J. Geom. Anal. 33, Art. 274, (2023)]).

Homogeneous hypersurfaces of the four-dimensional Thurston geometry ${\rm Sol_0^4}$

TL;DR

This work determines all homogeneous hypersurfaces in the four-dimensional Thurston geometry by classifying hypersurfaces with constant principal curvatures under constant angle functions. The authors develop the ambient geometry, derive the Gauss-Codazzi equations for hypersurfaces in this space, and identify four explicit model families with precise principal curvatures. They then prove that any homogeneous hypersurface in is congruent to one of these models, thus solving a standing problem in EI2. The results provide explicit parametrizations, curvature data, and orbit descriptions, advancing the understanding of isoparametric and homogeneous submanifolds in non-symmetric Thurston geometries.

Abstract

In this paper, we classify hypersurfaces with constant principal curvatures in the four-dimensional Thurston geometry under certain geometric conditions. As an application of the classification result, we give a complete classification of homogeneous hypersurfaces in , which solves a problem raised by Erjavec and Inoguchi (Problem 6.4 of [J. Geom. Anal. 33, Art. 274, (2023)]).

Paper Structure

This paper contains 8 sections, 15 theorems, 90 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The geometric structure on Sol40
  4. Hypersurfaces of Sol40
  5. Examples of homogeneous hypersurfaces of Sol40
  6. Proofs of Theorem 1.1 and Theorem 1.2
  7. Proof of Theorem 1.1
  8. Proof of Theorem 1.2

Key Result

Theorem 1.1

Let $M$ be a hypersurface of ${\rm Sol_0^4}$ with constant principal curvatures and constant angle functions $c$ and $d$. Then, up to isometries of ${\rm Sol_0^4}$, one of the following four cases occurs:

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Example 3.1
  • Proposition 3.1
  • proof
  • Example 3.2
  • Proposition 3.2
  • proof
  • Example 3.3
  • ...and 23 more