On versality for unfoldings of smooth section germs
type: Item
instance of: scholarly article
title: On versality for unfoldings of smooth section germs
zbMATH Open document ID: 0531.58011
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1982-01-01T00:00:00Z
review text: In recent years, the ''versality theorem'' for categories of unfoldings of smooth map germs relative to some equivalence relations has been proved (Martinet, Mather, Zakalyukin). 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 fields germs relative to coordinate transformations, there are examples for which the ''versality theorem'' cannot hold [see \textit{G. R. Belitskij,} Usp. Mat. Nauk 33, No.1(199), 95- 155 (1978; Zbl 0385.58007); translation from Russ. Math. Surv. 33, No.1, 107-177 (1978)]. In this paper, the author singles out a class of categories of unfoldings of smooth section germs of smooth vector bundles relative to various equivalence relations for which the ''versality theorem'' holds. As corollaries of the main theorem, the author shows the ''versality theorem'' for categories of unfoldings of smooth map germs (Martinet, Mather, Zakalyukin), and the category of G-unfoldings of G-invariant function germs relative to G-right equivalence [\textit{V. Poenaru}, Singularités \(C^{\infty}\) en présence de symétrie (Lect. Notes Math. 510) (1976; Zbl 0325.57008)]. Namely, the main theorem of this paper unifies these theorems. Moreover, it gives categories other than those stated above for which the ''versality theorem'' holds. For example, it is obtained that for the category of unfoldings of solutions of Poisson equations \(\Delta f=g\) the ''versality theorem'' holds.
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
Mathematics Subject Classification ID: 58H05
Mathematics Subject Classification ID: 58A20
zbMATH DE Number: 3842805
zbMATH Keywords: versality theorem
zbMATH Keywords: infinitesimal versality
zbMATH Keywords: category of unfoldings
zbMATH Keywords: Poisson equations
author: Shyuichi Izumiya
reviewed by: Masahisa Adachi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195184018
OpenAlex ID: W2036578980
cites work: A theory for imperfect bifurcation via singularity theory
cites work: Q4136232
cites work: Smooth functions invariant under the action of a compact Lie group
DOI: 10.2977/PRIMS/1195184018
description: scientific article; zbMATH DE number 3842805