Geometric and numerical methods for bonnet problems and surface construction
This PhD dissertation studies several important problems in differential geometry and its numerical applications. Its novel contributions to these areas include the following:
- solving Bonnet problems using the Cartan theory of moving frames, Cartan structure equations, and numerical analysis;
- studying surface duality in Riemannian geometries, in terms of a certain duality between mean curvature and the Hopf differential factor;
- determining sufficient geometric conditions for a surface to admit isothermic coordinates;
- constructing surfaces that admit isothermic coordinates; determining isothermic coordinate charts when starting from an arbitrary chart; implementing these methods numerically.
The moving frames and Cartan structure equations are written in terms of the first and second fundamental forms, and the Lax system is consequently reinterpreted; orthonormal moving frames are obtained as solutions to the Bonnet-Lax system, via numerical integration. Certain classifications of families of surfaces are studied in terms of the first and second fundamental forms, with respect to certain prescribed invariants. Numerical methods are applied to this theoretical framework in order to solve Bonnet problems, construct isothermic coordinate charts for surfaces that admit them, and construct dual surfaces (Christoffel transforms). Several visual examples are provided, as well as the corresponding numerical methods and code.