Smooth Stratified Vector Bundles and Obstructions to Their Orthonormal Frame Bundles

Abstract

Motivated by the example of the tangent bundle of a stratified space, which is no longer a vector bundle, we begin the construction of a general theory of smooth stratified vector bundles. We show that one can construct a frame bundle of a smooth stratified vector bundle in a canonical way, but that there are substantial obstructions to constructing an orthonormal frame bundle.