Nash \(C^ 1\)-embedding theorem for Carnot-Carathéodory metrics
type: Item
instance of: scholarly article
title: Nash \(C^ 1\)-embedding theorem for Carnot-Carathéodory metrics
zbMATH Open document ID: 0831.53022
DOI: 10.1016/0926-2245(95)00010-2
author: Giuseppina D'Ambra
published in: Differential Geometry and its Applications
publication date: 1996-02-12T00:00:00Z
review text: The Nash \(C^1\)-embedding theorem for Riemannian manifolds is generalized to contact Carnot-Caratheodory manifolds. Namely, it is proved the following: Let \(V\) and \(W\) be manifolds with contact structures \(S \subset T(V)\) and \(T \subset T(W)\) and let \(g\) and \(h\) be positive definite quadratic forms defined on \(S\) and \(T\) respectively, where \(T\) is globally represented as \(\text{Ker }\beta\) for some 1-form \(\beta\) on \(W\). Assume, moreover, that \(h\) agrees with the curvature form \(\omega = d\beta|_T\) in the sense that they together define a Hermitian structure on \(T\). If \(\dim W \geq 2\dim V + 3\) and \(V\) is compact then there exists a \(C^1\)-embedding \(f : V \to W\) which is contact and isometric, in the sense that \(Df\) sends \((S,g)\) isometrically onto \((T,h)\).
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 778515
zbMATH Keywords: Carnot-Caratheodory metric
zbMATH Keywords: isometric immersion
zbMATH Keywords: contact structure
reviewed by: R. Tribuzy
MaRDI profile type: Publication
full work available at URL: 0926-2245(95)00010-2
OpenAlex ID: W2132151362
Wikidata QID: Q115362571
cites work: TOPOLOGICAL CHARACTERIZATION OF STEIN MANIFOLDS OF DIMENSION >2
cites work: Some global properties of contact structures
cites work: Partial Differential Relations
cites work: Q4718347
cites work: Q3229047
cites work: On Carnot-Caratheodory metrics
cites work: \(C^1\) isometric imbeddings
cites work: Carnot-Carathéodory metrics and quasiisometries of symmetric spaces of rank 1
description: scientific article; zbMATH DE number 778515