Isometry of domains in \({\mathbb{R}}^ n\) and relative isometry of their boundaries
A domain D of \({\mathbb{R}}^ n\) is by definition an open and connected subset so that the metric of \({\mathbb{R}}^ n\) may serve D as its intrinsic metric. Herewith the intrinsic metric of a topological space M is the metric such that the distance between any two points is given by the infimum of the lengths of all rectifiable curves between these two points, and one deals with domains in \({\mathbb{R}}^ n\) whose intrinsic metric is continuously extendable to a metric on the closure of the domain. In particular, the restriction of this extended metric to the boundary of the domain D, denoted by \(^{\rho}D\), will be called the relative metric. For two domains D and D' in \({\mathbb{R}}^ n\) with boundaries S and S', respectively, a mapping \(f: S\to S'\) is said to be isometric in the relative metrics of the boundaries of D and D' if for any two points x and y of S it holds that \(^{\rho}D(f(x),f(y))=^{\rho}D'(x,y)\). Then the boundaries S and S' may also be called isometric in the relative metrics if there exists a surjective mapping \(f: S\to S'\) which is isometric in the relative metrics. Evidently, the boundaries of D and D' are isometric if the relative metrics of D and D' are isometric. However, it is an open problem to ask if the converse statement is true, namely, does isometry of the boundaries of domains in the relative metrics imply isometry of domains themselves ? In this paper the author treats this problem for bounded domains with piecewise-smooth boundary. As its preliminary fact, he proves the following theorem: Let D and D' be bounded domains on \({\mathbb{R}}^ n\), \(n\geq 2\), whose boundaries S and S' are (n-1)-dimensional differentiable manifolds of class \(C^ 1\) without boundary. If S and S' are isometric in the relative metrics, then D and D' are isometric. This theorem is then used to prove the following general theorem which states that, for bounded domains D and D' of \({\mathbb{R}}^ n\), \(n\geq 2\), with piecewise-smooth boundaries, if their boundaries are isometric in the relative metrics, then the domains are themselves isometric.
scientific article; zbMATH DE number 3921426