Convex isometric folding
type: Item
instance of: scholarly article
title: Convex isometric folding
zbMATH Open document ID: 0993.53009
DOI: 10.1155/S0161171200001198
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2000-09-28T00:00:00Z
full work available at URL: https://eudml.org/doc/49384
review text: Let \(M\) and \(N\) be \(C^\infty\) Riemannian manifolds of dimensions \(m\) and \(n\), respectively. A map \(\varphi :M\rightarrow N\) is said to be an isometric folding of \(M\) into \(N\) if for any piecewise geodesic path \(\gamma :J\rightarrow M\), the induced path \(\varphi _{\circ }\gamma :J\rightarrow N\) is a piecewise geodesic path of the same length. This definition is given by \textit{S. A. Robertson} [Proc. R. Soc. Edinb., Sect. A 79, 275-284 (1977; Zbl 0418.53016)]. The set of all isometric foldings \(\varphi :M\rightarrow N\) is denoted by \(\mathcal{J}(M,N).\) Let \(p:M\rightarrow N\) be a regular locally isometric covering and let \(G\) be the group of covering transformations of \(p\). An isometric folding \(\phi \in \mathcal {J}(M)\) is said to be \(p\)-invariant if for all \(g\in G\) and all \(x\in M\), \(p(\phi (x))=p(\phi (g,x))\). The article contains some confusion with \(\phi \) and \(\varphi \) as well as \(X\) and \(M\) here. The set of \(p\)-invariant isometric foldings is denoted by \(\mathcal{J}_{i}(M,p).\) Let \(\varphi \in \mathcal{J}(M,N)\) (resp. \(\varphi \in \mathcal{J}_{i}(M,p)\)); \(\varphi \) is a convex isometric folding if and only if \(\varphi (M)\) can be embedded as a convex set in \(\mathbb{R}^{n}\) (resp. \(\mathbb{R}^{m}\)). The set of all convex isometric foldings of \(M\) into \(N\) is denoted by \(\mathcal{C}(M,N)\). The set of all \(p\)-invariant convex isometric foldings of \(M\) into \(N\) is denoted by \(\mathcal{C}_{i}(M,p).\) It is known that if \(N\) is an \(n\)-smooth Riemannian manifold, \(p:M\rightarrow N\), its universal covering, then: 1. If \(G\) is the group of covering transformations of \(p\), then \(\mathcal{J}(N)\) is isomorphic as a semigroup to \(\mathcal{J}_{i}(M,p)/G\) [\textit{S. A. Robertson} and \textit{E. M. Elkholy}, Delta J. Sci. 8(2), 374-385 (1984)]. 2. If \(\varphi \in \mathcal{J}(N)\) is such that \(\varphi _{*}:\pi _{1}(N)\rightarrow \pi _{1}(N)\) is trivial, then the corresponding fo…
reviewed by: Ana Pereira do Vale
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 57N16
Mathematics Subject Classification ID: 51H10
zbMATH DE Number: 1455559
zbMATH Keywords: isometric folding
zbMATH Keywords: convex folding
zbMATH Keywords: regular covering
zbMATH Keywords: universal covering
zbMATH Keywords: fundamental region
author: E. M. El-Kholy
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1455559