Generalized splitting theorem for map-germs
type: Item
instance of: scholarly article
title: Generalized splitting theorem for map-germs
zbMATH Open document ID: 0545.58011
author: Fumio Ichikawa
published in: Proceedings of the Japan Academy. Series A
publication date: 1983-01-01T00:00:00Z
review text: Let \({\mathcal E}(n,p)\) denote the algebra of formal (power series) mappings \(({\mathbb{R}}^ n,0)\to({\mathbb{R}}^ p,0)\) (and similarly for \({\mathbb{C}})\), and let \({\mathfrak M}={\mathcal E}(n,1)\). Let \({\mathcal E}_ i(n,p)\subset {\mathcal E}(n,p)\) be the vector space of homogeneous polynomials of degree i. If \(f\in {\mathcal E}(n,p)\), then \(f=f_ k+f_{k+1}+...\) with \(f_ i\in {\mathcal E}_ i(n,p)\). Let \(G_ i\) be a complement linear subspace of \({\mathcal E}_ i(n,p)\cap {\mathfrak M}^ 2\cdot<\partial f_ k/\partial x>\) in \({\mathcal E}_ i(n,p)\). The claim is: There exists a formal diffeomorphism \(\phi:({\mathbb{R}}^ n,0)\to({\mathbb{R}}^ n,0)\) such that \(f\circ \phi =f_ k+g_{k+1}+g_{k+2}+...\) with \(g_ i\in G_ i\). This contains the Morse lemma, the splitting lemma and a parametrized version contains Whitney's fold and cusp normal forms (in the formal category). This is an announcement without proofs.
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
zbMATH DE Number: 3868268
zbMATH Keywords: Morse lemma
zbMATH Keywords: splitting lemma
zbMATH Keywords: fold and cusp normal forms
MaRDI profile type: Publication
full work available at URL: pjaa.59.422
OpenAlex ID: W2021745650
cites work: On the order of determination of a finitely determined germ
cites work: A note on the order of determination of a finitely determined germ
cites work: Stability of \(C^ \infty\) mappings. III: Finitely determined map germs
cites work: On the determinacy of smooth map-germs
cites work: Finite Determinacy of Smooth Map-Germs
cites work: Stability of unfoldings
DOI: 10.3792/PJAA.59.422
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description: scientific article; zbMATH DE number 3868268