Genuine deformations and submanifolds
type: Item
instance of: scholarly article
title: Genuine deformations and submanifolds
zbMATH Open document ID: 1068.58005
author: Marcos Dajczer
author: Luis A. Florit
published in: Communications in Analysis and Geometry
publication date: 2005-03-18T00:00:00Z
review text: The authors introduce the concept of genuine isometric deformation of a Euclidean submanifold and they describe the geometric structure of submanifolds that admit deformations of this kind. Definition: An isometric immersion \(\widehat{f}:M^n\mapsto\mathbb R^{n+q}\) is a genuine deformation of a given isometric immersion \(f:M^n\mapsto\mathbb R^{n+p}\) if there is no open subset \(U\subset M^n\) along which the restictions \(f_{| U}\) and \(\widehat{f}_{| U}\) extend isometrically. The authors prove that any pair of submanifolds in low codimension determined by a genuine transformation is mutually ruled (with the same rullings) and they give sharp estimate for the dimension of the rullings. In addition they show that the relation discussed in the sequel between the normal bundles and second fundamental forms that exists for any pair of mutually real submanifolds must satisfy strong additional conditions. The main result of the paper is as follows. Theorem: Let \(\widehat{f}:M^n\mapsto\mathbb R^{n+q}\) be a genuine deformation of \(f:M^n\mapsto\mathbb R^{n+p}\) with \(p+q<n\) and \(\min\{p,q\}\leq 6\). Then \(f\) and \(\widehat{f}\) are mutually \(D^d\)-ruled along each connected component of an open dense subset of \(M^n\) with \(d\geq n-p-q+3l\) unless \(\min\{p,q\}=6\) and \(l=0\) in which case \(d=n-p-q-1\) and the bundle isometry \(T:L^l\mapsto L^l\) satisfies two conditions: (\(C_1\)) The isometry \(T\) is parallel and preserves second fundamental forms (\(C_2\)). The subbundles \(L\) and \(\widehat{L}\) are parallel along \(D\) in the normal connections.
reviewed by: Jan Kurek
Mathematics Subject Classification ID: 58D99
Mathematics Subject Classification ID: 58H99
Mathematics Subject Classification ID: 53C99
zbMATH DE Number: 2147076
MaRDI profile type: Publication
full work available at URL: cag.2004.v12.n5.a6
OpenAlex ID: W2067638626
Wikidata QID: Q125620064
DOI: 10.4310/CAG.2004.V12.N5.A6
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description: scientific article; zbMATH DE number 2147076