Thickness Formula and C1-Compactness for C1,1 Riemannian Submanifolds
O. C. Durumeric
Abstract
We prove a formula for the normal injectivity radius(thickness)i(K,M)for C1,1 compact submanifolds Kk of complete Riemannian manifolds Mn in terms of geometric focal distance and double critical points. We also prove the C1 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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