Compact Bonnet Pairs: isometric tori with the same curvatures

TL;DR

This paper constructs the first known examples of compact Bonnet pairs, demonstrating that metric and mean curvature do not uniquely determine a smooth compact immersion, and proves these tori are real analytic.

Contribution

It explicitly constructs the first compact Bonnet pairs and proves their real analyticity, resolving longstanding open problems in differential geometry.

Findings

First examples of compact Bonnet pairs.

Proved the real analyticity of these tori.

Showed metric and mean curvature do not determine unique immersion.

Abstract

We explicitly construct a pair of immersed tori in three dimensional Euclidean space that are related by a mean curvature preserving isometry. These Bonnet pair tori are the first examples of compact Bonnet pairs. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique smooth compact immersion. Moreover, we prove these isometric tori are real analytic. This resolves a second longstanding open problem on whether real analyticity of the metric already determines a unique compact immersion. Our construction uses the relationship between Bonnet pairs and isothermic surfaces. The Bonnet pair tori arise as conformal transformations of an isothermic torus with one family of planar curvature lines. We classify such isothermic tori in our companion paper (arXiv:2312.14956). The above approach stems from computational investigations of a 5x7 quad decomposition of a torus using a discrete differential geometric analog of isothermic surfaces and Bonnet pairs.