Conformal submanifold geometry I-III

By F. Burstall, D. Calderbank. 2010

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In Part I, we develop the notions of a Moebius structure and a conformal Cartan geometry, establish an equivalence between them; we use them in Part II to study submanifolds of conformal manifolds in arbitrary dimension and codimension. We obtain Gauss-Codazzi-Ricci equations and a conformal Bonnet theorem characterizing immersed submanifolds of the conformal n-sphere. These methods are applied in Part III to study constrained Willmore surfaces, isothermic surfaces, Guichard surfaces and conformally-flat submanifolds with flat normal bundle, and their spectral deformations, in arbitrary codimension. The high point of these applications is a unified theory of Moebius-flat submanifolds, which include Guichard surfaces and conformally flat hypersurfaces.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b1ac08ba86c03a02f88b51f6381e066f3ad37661