Quadratic Differentials, Quaternionic Forms, and Surfaces

Details Quadratic Differentials, Quaternionic Forms, and Surfaces Quadratic Differentials, Quaternionic Forms, and Surfaces

1997-12-21

arxiv.org/abs/dg-ga/9712011v2

Mathematics

Differential Geometry

LaTeX. Stylistic changes and spelling mistakes corrected. Definitions 2 and 3, and Remark 4 are streamlined

Scientific paper

Global isothermic immersions are defined and studied with the aid of a connection between quadratic differentials and immersions. The applications are two problems stemming from the fundamental question: how much data is needed to identify a surface immersion (Christoffel's problem) or its shape (Bonnet's problem). A short complete solution of Christoffel's problem, including closed surfaces, is given. It is shown that every immersion of an oriented closed surface (genus $\neq$ 1) is uniquely determined up to similitude by its conformal class and the tangent planes map. A classification of all generic Bonnet surfaces follows from a series of papers by Bonnet, Cartan, and Chern. The existence of a new class of Bonnet surfaces is shown here. The understanding of this class is necessary in order to study the rigidity of closed surfaces.

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