A conformal differential invariant and the conformal rigidity of hypersurfaces

Geometric Rigidity of Conformal Matrices

We provide a geometric rigidity estimate à la Friesecke-James-Müller for conformal matrices. Namely, we replace SO(n) by a arbitrary compact subset of conformal matrices, bounded away from 0 and invariant under SO(n), and rigid motions by Möbius transformations.

On Isometric and Conformal Rigidity of Submanifolds

Let f, g : Mn → Rn+d be two immersions of an n-dimensional differentiable manifold into Euclidean space. That g is conformal (isometric) to f means that the metrics induced on Mn by f and g are conformal (isometric). We say that f is conformally (isometrically) rigid if given any other conformal (isometric) immersion g there exists a conformal (isometric) diffeomorphism Υ from an open subset of...

Eta Invariant and Conformal Cobordism

In this note we study the problem of conformally flat structures bounding conformally flat structures and show that the eta invariants give obstructions. These lead us to the definition of an Abelian group, the conformal cobordism group, which classifies the conformally flat structures according to whether they bound conformally flat structures in a conformally invariant way. The eta invariant ...

ar X iv : m at h / 98 06 00 1 v 1 [ m at h . D G ] 3 0 M ay 1 99 8 A CONFORMAL DIFFERENTIAL INVARIANT AND THE CONFORMAL RIGIDITY OF HYPERSURFACES

For a hypersurface V n−1 of a conformal space, we introduce a conformal differential invariant I = h 2 g , where g and h are the first and the second fundamental forms of V n−1 connected by the apolarity condition. This invariant is called the conformal quadratic element of V n−1. The solution of the problem of conformal rigidity is presented in the framework of conformal differential geometry ...

Rigidity and Inflexibility in Conformal Dynamics