The differential properties of one class of surfaces in Euclidean space

Abstract

We consider n-dimensional smooth surfaces of class ℰ¹ in the Euclidean space of dimension n + m satisfying the following condition. Given two distinct points of the surface, the surface normals at these points either are disjoint or meet at the distance from both of these points bounded below by some fixed positive constant. We establish that every surface of this type carries in a neighborhood of each point a parametrization with bounded second order generalized derivatives in the sense of Sobolev. The proof is based on using geometric properties of the surfaces of this form and on the proposition that establishes sufficient conditions for the existence of bounded second order generalized derivatives of an arbitrary real function. In the Appendix we prove an analog of this lemma in the case of derivatives of arbitrary order.