Compatibility equations for isometric embeddings of Riemannian manifolds
type: Item
instance of: scholarly article
title: Compatibility equations for isometric embeddings of Riemannian manifolds
zbMATH Open document ID: 0813.53006
author: Chung-Ki Cho
author: Chong-Kyu Han
published in: Rocky Mountain Journal of Mathematics
publication date: 1994-07-04T00:00:00Z
review text: This paper considers local isometric immersions of Riemannian manifolds \(M\) into Euclidean spaces. Through a prolongation of the underlying PDE it constructs compatibility equations by a method due to A. Finzi, and discusses some of their geometric consequences. Some specific results: If a \(C^ 2\)-isometric embedding of \(M\) is \(C^ \infty\) except at a hypersurface \(H\) of \(M\), then \(H\) must be asymptotic (Corollary 11). Any two isometric embeddings contained in a \(C^ 3\)-neighborhood of an infinitesimally rigid elliptic (in the sense of N. Tanaka) isometric embedding are congruent (Theorem 13). Any two real analytic elliptic isometric embeddings of \(M\) coinciding along a hypersurface \(H\) of \(M\) must coincide on a neighborhood of \(H\) (Theorem 18).
reviewed by: Eberhard Teufel
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 599723
zbMATH Keywords: rigidity
zbMATH Keywords: elliptic PDE
zbMATH Keywords: isometric immersions
zbMATH Keywords: elliptic isometric embeddings
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072490
OpenAlex ID: W1997775202
Wikidata QID: Q115239921
cites work: The Gauss equations and rigidity of isometric embeddings
cites work: Q3997815
cites work: Characteristics and existence of isometric embeddings
cites work: Q5785236
cites work: Differential Systems and Isometric Embeddings.(AM-114)
cites work: Regularity of certain rigid isometric immersions of \(n\)-dimensional Riemannian manifolds into \({\mathbb{R}}^{n+1}\)
cites work: Regularity of isometric immersions of positively curved Riemannian manifolds and its analogy with CR geometry
cites work: Q5679391
cites work: Q3968082
cites work: Q3998623
cites work: Q5565587
cites work: Rigidity for isometric imbeddings
cites work: Q5685459
cites work: Q3715336
cites work: Q3209062
cites work: On the Gauss equation of an isometric immersion
cites work: Rigidity for elliptic isometric imbeddings
cites work: On characteristic hypersurfaces of submanifolds in Euclidean space
cites work: On Infinitesimal Isometric Deformations
cites work: Q3938154
DOI: 10.1216/RMJM/1181072490
Recommended article: Q4011217
Recommended article: Relative isometric embeddings of Riemannian manifolds
Recommended article: Q3408353
Recommended article: Isometric embeddings of Riemannian and pseudo-Riemannian manifolds.
Recommended article: Isometries of complemented sub-Riemannian manifolds
Recommended article: On the perturbation problem associated to isometric embeddings of Riemannian manifolds
Recommended article: Q4289755
Recommended article: Isometric embedding of a compact Riemannian manifold into sphere
Recommended article: Q4727007
Recommended article: Isometric immersions of Riemannian manifolds
description: scientific article; zbMATH DE number 599723