The Bonnet Problem

Abstract

This chapter presents the Bonnet Problem, which asks whether an immersion of a surface x: M → R ³ admits a Bonnet mate, which is another noncongruent immersion $\tilde{\mathbf{x}}: M \rightarrow \mathbf{R}^{3}$ with the same induced metric and the same mean curvature at each point. Any immersion with constant mean curvature admits a 1-parameter family of Bonnet mates, all noncongruent to each other. These are its associates. The problem is thus to determine whether an immersion with nonconstant mean curvature has a Bonnet mate. The answer for an umbilic free immersion is whether it is isothermic or not. If it is nonisothermic, then it possesses a unique Bonnet mate. We believe that this is a new result. If it is isothermic, then only in special cases will it have a Bonnet mate, and if it does, it has a 1-parameter family of mates, similar to the CMC case. Such immersions are called proper Bonnet. A brief introduction to the notion of G-deformation is used to derive the KPP Bonnet pair construction of Kamberov, Pedit, and Pinkall. We state and prove a new result on proper Bonnet immersions that implies results of Cartan, Bonnet, Chern, and Lawson-Tribuzy. The chapter concludes with a summary of Cartan’s classification of proper Bonnet immersions.