Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geodesic-preserving bijections of the Thurston geometries

Ryan Dickmann, Palani Lideros, Akash Narayanan

TL;DR

This work completes the classification of geodesic-preserving bijections across the Thurston geometries not fully settled by Jeffers. It shows that for the product geometries $oldsymbol{oldsymbol{H}^2 imesoldsymbol{R}}$ and $oldsymbol{oldsymbol{S}^2 imesoldsymbol{R}}$, such bijections are isometries composed with an affine map in the $oldsymbol{R}$-coordinate, while in the remaining three geometries $ ilde{SL}_2(oldsymbol{R})$, Nil, and Sol, geodesic-preserving bijections are outright isometries. The arguments combine a general result that totally geodesic subsets are submanifolds under a mild geometric condition with geometry-specific analyses of geodesic types (horizontal/vertical/slant), projections (e.g., to $oldsymbol{H}^2$ or $oldsymbol{R}^2$), and constant-curvature curves. The cylinder analysis plays a crucial role in the $oldsymbol{S^2 imesoldsymbol{R}}$ case, while constant-curvature-curve preservation drives the $ ilde{SL}_2(oldsymbol{R})$ and Nil classifications, and hyperbolic-plane geometry underpins the Sol case. Overall, the results establish rigidity of geodesic-preserving bijections and complete the symmetry classification for these Thurston geometries.

Abstract

We completely classify the bijections of the Thurston geometries that preserve geodesics as sets. For Riemannian manifolds that satisfy a certain technical condition, we prove that a totally geodesic subset is a submanifold. We also classify the geodesic-preserving bijections of the Euclidean cylinder $\mathbb{S}^1 \times \mathbb{R}$ and the bijections of the hyperbolic plane $\mathbb{H}^2$ that preserve constant curvature curves.

Geodesic-preserving bijections of the Thurston geometries

TL;DR

This work completes the classification of geodesic-preserving bijections across the Thurston geometries not fully settled by Jeffers. It shows that for the product geometries and , such bijections are isometries composed with an affine map in the -coordinate, while in the remaining three geometries , Nil, and Sol, geodesic-preserving bijections are outright isometries. The arguments combine a general result that totally geodesic subsets are submanifolds under a mild geometric condition with geometry-specific analyses of geodesic types (horizontal/vertical/slant), projections (e.g., to or ), and constant-curvature curves. The cylinder analysis plays a crucial role in the case, while constant-curvature-curve preservation drives the and Nil classifications, and hyperbolic-plane geometry underpins the Sol case. Overall, the results establish rigidity of geodesic-preserving bijections and complete the symmetry classification for these Thurston geometries.

Abstract

We completely classify the bijections of the Thurston geometries that preserve geodesics as sets. For Riemannian manifolds that satisfy a certain technical condition, we prove that a totally geodesic subset is a submanifold. We also classify the geodesic-preserving bijections of the Euclidean cylinder and the bijections of the hyperbolic plane that preserve constant curvature curves.

Paper Structure

This paper contains 16 sections, 27 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Totally geodesic subsets are submanifolds
  3. Geodesic-preserving bijections of H2 times R
  4. Geodesic-preserving bijections of S2 times R
  5. Geodesic-preserving bijections of the cylinder
  6. Proof of the S2 times R case
  7. Geodesic-preserving bijections of the universal cover of SL2R
  8. Background
  9. Constant curvature curves
  10. Classification of geodesic-preserving bijections
  11. Geodesic-preserving bijections of Nil
  12. Background
  13. Classification of geodesic-preserving bijections
  14. Geodesic-preserving bijections of Sol
  15. Background
  16. ...and 1 more sections

Key Result

Proposition 1.1

Let $M$ be a Riemannian manifold with the property that there is an $\varepsilon > 0$ such that the intersection of a geodesic and an open $\varepsilon$-ball is either empty or a single geodesic segment. Then any totally geodesic subset is an embedded submanifold.

Figures (1)

  • Figure 1: For classifying the totally geodesic subsets of $\mathbb{H}^2 \times \mathbb{R}$, the chosen geodesic from $p(x)$ to $p(y)$ intersects $p(\gamma_r)$ only for $r$ in a bounded range. The $p(\gamma_r)$ for integer $r$ are shown.

Theorems & Definitions (52)

  • Proposition 1.1: Totally geodesic subsets are submanifolds
  • Remark
  • proof : Proof of Proposition \ref{['prop: fundlemma']}
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 42 more