Genuine rigidity of Euclidean submanifolds in codimension two
type: Item
instance of: scholarly article
title: Genuine rigidity of Euclidean submanifolds in codimension two
zbMATH Open document ID: 1076.53022
DOI: 10.1023/B:GEOM.0000033846.63094.46
author: Marcos Dajczer
author: Luis A. Florit
published in: Geometriae Dedicata
publication date: 2005-04-07T00:00:00Z
review text: An isometric deformation of a Euclidean submanifold is called genuine if the submanifold cannot be included into a submanifold of larger dimension in such a way that the deformation of the former is given by an isometric deformation of the latter. The submanifold is said to be genuinely rigid if it has no genuine deformation. In this paper, the authors study the deformation problem in codimension two for the classes of elliptic and parabolic submanifolds. In spite of having a second fundamental form as degenerate as possible without being flat, i.e., the Gauss map has rank two everywhere, their main result is as follows: If \(f:M^n\to \mathbb R^{n+2}\) is a nowhere surface-like elliptic submanifold that admits no local isometric immersion in \(\mathbb R^{n+1}\), then \(f\) is genuinely rigid. Moreover, an additional unexpected deformation phenomenon for elliptic submanifolds carrying a Kähler structure is described.
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 2153904
zbMATH Keywords: isometric immersion of rank two
zbMATH Keywords: isometric rigidity
zbMATH Keywords: isometric deformation
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2153904