On Versality for Unfoldlngs of Smooth Section Germs

In this article, we consider unfoldings of smooth section germs. The unfolding, which is introduced by Rene Thorn, is an important notion to describe any situations in which qualitative picture of the object with a change of the parameters on which the object depends. The idea is an analogue to the deformation theory of singularities of complex varieties for real smooth cases. Hence, it can be thought as smooth families of section germs. It turns out in many cases the study of all possible unfoldings leads to that of a single one, from which all others can be obtained. Such an unfolding, in some sense the largest one, should give all the essentially distinct bifurcation with respect to given equivalence relation; it is the versal unfolding. In recent years, "the versality theorem" for categories of unfoldings of smooth map germs relative to some equivalence relations have been proved ([3]5 [4], [8]). Now, we say that the versality theorem holds if the algebraic notion of "infinitesimal versality" is the sufficient condition of the notion of "versality". But as the category of unfoldings of smooth vector field germs relative to coordinate transformations, there are examples for which "the versality theorem" cannot hold ([!]). In this paper, we will single out the class of categories of unfoldings of smooth section germs of smooth vector bundles relative to various equivalence relations for which "the versality theorem" hold. As applications of the main theorem, we have "versality theorems" for