A rigidity theorem for higher codimensions
type: Item
instance of: scholarly article
title: A rigidity theorem for higher codimensions
zbMATH Open document ID: 0579.53006
DOI: 10.1007/BF01458592
author: Marcos Dajczer
author: Manfredo P. do Carmo
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164117
review text: Let \(f: M^ n\to R^{n+k}\) be an isometric immersion of an n- dimensional, connected, Riemannian manifold \(M^ n\) into the \((n+k)\)- Euclidean space \(R^{n+k}\), and let \(B=\{p\in M\); \(\nu (p)=n\}\) be the set of totally geodesic points of f; here \(\nu\) is the nullity of the second fundamental form of f. Assume that M is compact and that M-B is connected. It is known that if \(k=1\), f is rigid (unique up to rigid motions). Here the following generalization of this theorem is proved. Let \(\nu_ s(p)\), \(p\in M\), be the maximum of the nullities of the projections of the second fundamental form onto s-dimensional subspaces of the normal space at p (for \(k=1\), \(\nu_ s=\nu)\). For \(k>1\), assume, in addition to the above conditions, that \(\nu_ s(p)\leq n-(2s+1)\), \(1\leq s\leq k-1\). Then if \(k\leq 5\), f is rigid.
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 3927809
zbMATH Keywords: rigidity
zbMATH Keywords: totally geodesic points
zbMATH Keywords: second fundamental form
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3927809