Generalized Hestenes’ Lemma and extension of functions

Abstract

Suppose we have an m-jet field on V C Rn which is a Whitney field on the nonsingular part M of V We show that, under certain hypotheses about the relationship between geodesic and euclidean distance on V, if the field is flat enough at the singular part S then it is a Whitney field on V (the order of flatness required depends on the coefficients in the hypotheses). These hypotheses are satisfied when V is subanalytic. In Section TI) we show that a C2 function f on M can be extended to one on V if the diSerential dy goes to 0 faster than the order of divergence of the principal curvatures of M and if the first covariant derivative of df is sufficiently flat. For the general case of cm functions with m > 2, we give a similar result for codim M = 1 in Section II