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Uniqueness of the Bonnet problem in Thurston geometries

José S. Santiago

TL;DR

Thurston geometries extend the classical Bonnet problem to homogeneous 3-manifolds, with the main results showing uniqueness of Bonnet mates in most cases and explicit classifications for when continuous deformations preserving principal curvatures occur. In the product spaces $ ext{E}( kappa,0)$ these deformations form an $ ext{S}^1$-family only for minimal or constant-principal-curvature immersions; for $ au eq0$ they occur under constant principal curvatures, with delicate counts for positive versus negative mates. For Sol$_3$, Bonnet mates are characterized by two first-order PDEs in angle data, and except when the left-invariant Gauss map is constant, there are finitely many (at most seven) noncongruent mates, thereby solving a Chern-type problem in Sol$_3$. Together these results connect with invariance under one-parameter isometry groups and extrinsically homogeneous (isoparametric) surfaces, providing both new examples and a comprehensive framework for the Bonnet problem in non-Euclidean homogeneous geometries.

Abstract

We study the Bonnet problem in Bianchi--Cartan--Vrănceanu spaces and in $\mathrm{Sol}_3$. Our main contribution is to establish the uniqueness of Bonnet mates, which leads us to address the problem of determining when an isometric immersion can be continuously deformed through isometric immersions that preserve the principal curvatures -- a question originally posed in $\mathbb{R}^3$ by Chern~\cite{Chern}. For Bianchi--Cartan--Vrănceanu spaces, we complete the local classification of Bonnet pairs by studying the uniqueness of the results obtained by Gálvez, Martínez and Mira~\cite{GMM}, and we provide new examples of Bonnet mates that were not previously considered. In particular, we prove that the aforesaid continuous deformations only exist for minimal surfaces in the product spaces $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$ and otherwise only for surfaces with constant principal curvatures. In the case of $\mathrm{Sol}_3$, we give a characterization of Bonnet mates via a system of two differential equations, addressing a problem proposed in~\cite{GMM}. We conclude that the only surfaces admitting continuous isometric deformations that preserve the principal curvatures in $\mathrm{Sol}_3$ are those with constant left-invariant Gauss map.

Uniqueness of the Bonnet problem in Thurston geometries

TL;DR

Thurston geometries extend the classical Bonnet problem to homogeneous 3-manifolds, with the main results showing uniqueness of Bonnet mates in most cases and explicit classifications for when continuous deformations preserving principal curvatures occur. In the product spaces these deformations form an -family only for minimal or constant-principal-curvature immersions; for they occur under constant principal curvatures, with delicate counts for positive versus negative mates. For Sol, Bonnet mates are characterized by two first-order PDEs in angle data, and except when the left-invariant Gauss map is constant, there are finitely many (at most seven) noncongruent mates, thereby solving a Chern-type problem in Sol. Together these results connect with invariance under one-parameter isometry groups and extrinsically homogeneous (isoparametric) surfaces, providing both new examples and a comprehensive framework for the Bonnet problem in non-Euclidean homogeneous geometries.

Abstract

We study the Bonnet problem in Bianchi--Cartan--Vrănceanu spaces and in . Our main contribution is to establish the uniqueness of Bonnet mates, which leads us to address the problem of determining when an isometric immersion can be continuously deformed through isometric immersions that preserve the principal curvatures -- a question originally posed in by Chern~\cite{Chern}. For Bianchi--Cartan--Vrănceanu spaces, we complete the local classification of Bonnet pairs by studying the uniqueness of the results obtained by Gálvez, Martínez and Mira~\cite{GMM}, and we provide new examples of Bonnet mates that were not previously considered. In particular, we prove that the aforesaid continuous deformations only exist for minimal surfaces in the product spaces and and otherwise only for surfaces with constant principal curvatures. In the case of , we give a characterization of Bonnet mates via a system of two differential equations, addressing a problem proposed in~\cite{GMM}. We conclude that the only surfaces admitting continuous isometric deformations that preserve the principal curvatures in are those with constant left-invariant Gauss map.

Paper Structure

This paper contains 14 sections, 21 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Thurston geometries
  3. Bianchi-Cartan-Vrănceanu spaces
  4. The compatibility equations in $\mathbb{E}(\kappa,\tau)$
  5. $\mathrm{Sol}_3$-type geometries
  6. The compatibility equations in $\mathrm{Sol}_3$.
  7. Bonnet Mates
  8. Isometries of invariant surfaces
  9. The Bonnet problem in Bianchi-Cartan-Vrănceanu spaces
  10. The Bonnet problem in product spaces $\mathbb{E}(\kappa,0)$
  11. The Bonnet problem in $\mathbb{E}(\kappa,\tau)$ spaces
  12. Uniqueness of the Bonnet problem: positive case
  13. Uniqueness of the Bonnet problem: negative case
  14. The Bonnet problem in $\mathrm{Sol}_3$

Key Result

Proposition 2.2

Let $(\Sigma, \mathrm{d} s^2, J)$ be a simply connected oriented Riemannian surface. Let $S: T\Sigma \rightarrow T\Sigma$ be a field of self-adjoint endomorphisms, $T_3$ a vector field on $\Sigma$ and $\nu_3$ a function on $\Sigma$. Then there exists an isometric immersion $f: \Sigma \rightarrow \ma where $\nabla$ denotes the Levi-Civita connection of $\Sigma$. If these conditions hold, then the i

Theorems & Definitions (49)

  • Remark 2.1
  • Proposition 2.2: Dan07Dan09
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • Remark 2.7
  • Lemma 2.8
  • proof
  • ...and 39 more