The geometry of immersions. I

Abstract

Introduction. For the past twenty-five years much of global differential geometry and topology has centered around what we now call differential topology. Much emphasis has been placed on those mathemathical ideas (vector bundles, their characteristic classes, and general position arguments) which depend only on the differentiable structure of manifolds and maps. In particular the work of Whitney and Thorn (see Levine [1] and Whitney [l]-[6]) belong to that domain. Our principal aim in this paper is to show that the tools of differential topology can be suitably modified to become applicable to differential geometry. Our second goal is to use these tools to investigate some "higher-order" problems in differential geometry (that is, problems involving derivatives of order greater than 1).