Uniqueness of the Bonnet problem in Thurston geometries

TL;DR

This paper investigates the Bonnet problem in specific 3D geometries, establishing the uniqueness of Bonnet pairs, classifying when surfaces can be deformed isometrically while preserving principal curvatures, and providing new examples and characterizations.

Contribution

It advances the understanding of Bonnet pairs in Bianchi--Cartan--Vre2nceanu spaces and ol_3, including classification, new examples, and differential equation characterizations.

Findings

Continuous deformations only for minimal surfaces in ^2b7d7R and H^2b7d7R

Bonnet mates characterized by differential equations in ol_3

Surfaces with constant principal curvatures admit deformations

Abstract

We study the Bonnet problem in Bianchi--Cartan--Vr\u{a}nceanu 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--Vr\u{a}nceanu spaces, we complete the local classification of Bonnet pairs by studying the uniqueness of the results obtained by G\'alvez, Mart\'inez 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.