Thickness Formula and C^1-Compactness for C^{1,1} Riemannian Submanifolds

Details Thickness Formula and C^1-Compactness for C^{1,1} Riemannian Submanifolds Thickness Formula and C^1-Compactness for C^{1,1} Riemannian Submanifolds

2002-04-03

arxiv.org/abs/math/0204050v1

Mathematics

Differential Geometry

Scientific paper

We prove a formula for the normal injectivity radius(thickness)i(K,M)for C^{1,1} compact submanifolds K^k of complete Riemannian manifolds M^n in terms of geometric focal distance and double critical points. We also prove the C^1 compactness of the set of all compact submanifolds K contained in a compact subset D of a fixed manifold M with i(K,M) bounded away from 0. This is an extrinsic and isometric embeddindg type theorem related to Gromov's compactness theorem. This yields the existence of thickest submanifolds in an isotopy class where K is not necessarily connected.

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