Convex isometric folding (Q1567239)

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 folding \(\psi \in \mathcal{J}_{i}(M,p)\) maps each fiber of \(p\) to a single point [\textit{E. M. Elkholy}, ``Isometric and topological folding of manifolds'', Ph.D. thesis, Southampton University, England (1982)]. 3. If \(\varphi \in \mathcal{J}(N)\) is such that \(\varphi _{*}:\pi _{1}(N)\rightarrow \pi _{1}(N)\) is trivial, and \(N\) is a compact \(2\)-manifold then \(\frac{\text{Vol}(N)}{\text{Vol}(\varphi (N))}=\frac{\text{Vol}(F)}{\text{Vol}(\psi (F))}\) where \(F\) is a fundamental region \(G\) in \(M\). \textit{E. M. Elkholy} and \textit{A. E. Al-Ahmady} [Proc. Conf. Oper. Res. and Math. Methods, Bull. Fac. Sc. Alex. A 26 54-66 (1977)]. In this article the author proves that if \(C(N)\neq \emptyset \), then Theorem 1 is valid for \(\mathcal{C}(N)\) and \(\mathcal{C}_{i}(M,p)\), i.e., \(\mathcal{C}(N)\) is isomorphic as a semigroup to \(\mathcal{C}_{i}(M,p)/G\), and studies what happens regarding Theorems 2 and 3 for these sets. The author also proves that: if \(N\) is a compact orientable \(2\)-manifold and its universal covering space is (\(\mathbb{R}^{2},p)\) (respectively \((M,p)\)) and if \(\varphi \in \mathcal{C}(N)\) and \(\psi \in \mathcal{C}_{i}(\mathbb{R}^{2},p)\) (respectively \(\mathcal{C}_{i}(M,p)\)) then for all \(x,y\in \mathbb{R}^{2}\) (respectively \(M\)) \(d(\psi (x),\psi (x))\leq \Delta \), where \(\Delta \) is the radius of a fundamental region for the covering space. The author also proves that the infimum of the ratio \(\frac{\text{Vol}(N)}{\text{Vol}(\varphi (N))}\), where \(N\) is a compact \(2\)-manifold over all convex isometric foldings \(\varphi \in \mathcal{C}(N)\) of degree zero, is \(4\).