Trivializations of stratified spaces with bounded differential

Summary

Abstract We study the existence of trivializations of a riemannian stratified space whose differential has bounded norm giving sufficient conditions. We also prove some consequences of this property, the most meaningful of which is local finiteness of volume.

INTRODUCTION

This paper is the development of some considerations already contained in [F3] about trivializations of stratified spaces. It is well known from Thorn's isotopy lemmas that a b-regular (in the sense of Whitney) subset W of a smooth manifold is locally trivial in two senses : in one sense W is locally homeomorphic to the product of a smooth manifold with a stratified set, in the other the homeomorphism is with a mapping cylinder. Moreover, these homeomorphims are diffeomorphims on the strata. The abstract counterpart of these sets are the abstract or Thorn-Mather stratifications (which we call stratified spaces). If we consider on a stratified space W a collection of riemannian metrics g = {gx}, one for any stratum (in the embedded case g should be the restriction of a riemannian metric on the ambient manifold), it makes sense to ask if there exist trivializations such that the norm of the differential is bounded (D-bounded trivializations). We shall see that, in the mapping cylinder case, this has as a consequence that our space has locally finite volume. We point out that a b-regular stratified set has not in general locally finite volume, as shown in [F1], and that the positive results about finite volume given in [F] and [B] are based on the boundedness of the derivatives of a trivialization.