Thickness Formula and C-compactness for C Riemannian Submanifolds
نویسنده
چکیده
The properties of normal injectivity radius i(K, M) (thickness), of C submanifolds K of complete Riemannian manifolds M are studied. We introduce the notion of geometric focal distance for C submanifolds by using metric balls. A formula for i(K, M) in terms of the double critical points and the geometric focal distance is proved. The thickness of knots and ideal knots relate to the study of DNA molecules and other knotted polymers. We prove that the set of all C submanifolds K of a fixed manifold M contained in a compact subset D ⊂ M and i(K, M) ≥ c > 0 is C−compact and this collection has finitely many diffeomorphism and isotopy types. Estimates on upper bounds for the number of such types are constructible, and we calculate them for submanifolds of R. C−compactness is related to Gromov’s compactness theorem, but it is an extrinsic and isometric embedding type theorem.
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