Conformal submanifold geometry I-III

Abstract

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.

74 pages

References in corpus (4)

• Constrained Willmore Surfaces
• Ricci-corrected derivatives and invariant differential operators
• The Ribaucour transformation in Lie sphere geometry
• Bäcklund Transformations and Loop Group Actions

Cited by in corpus (25)

• Carrollian manifolds and null infinity: A view from Cartan geometry
• Asymptotic Shear and the Intrinsic Conformal Geometry of Null-Infinity
• Tractor geometry of asymptotically flat spacetimes
• Dressing transformations of constrained Willmore surfaces
• Twistors, 4-symmetric spaces and integrable systems
• Quantization of the conformal arclength functional on space curves
• Projective geometry and the quaternionic Feix-Kaledin construction
• Curvilinear coordinates on generic conformally flat hypersurfaces and constant curvature 2-metrics
• Lie applicable surfaces
• Ribaucour coordinates
• Generalised Bianchi permutability for isothermic surfaces
• Discrete cyclic systems and circle congruences
• Geodesics and Submanifold Structures in Conformal Geometry
• Notes on transformations in integrable geometry
• Discrete constant mean curvature surfaces via conserved quantities
• Lie Algebroid Invariants for Subgeometry
• Formal conserved quantities for isothermic surfaces
• A Characterisation of Smooth Maps into a Homogeneous Space
• Polynomial conserved quantities for constrained Willmore surfaces
• Transformations of harmonic bundles and Willmore surfaces
• Möbius-flat hypersurfaces in projective space
• Quartic differentials and harmonic maps in conformal surface geometry
• Generic conformally flat hypersurfaces and surfaces in 3-sphere
• A note on Ribaucour transformations in Lie sphere geometry
• Darboux and Calapso transforms of meromorphically isothermic surfaces