On the unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics

The article contains the results of the author's recent investigations of the rigidity problems of domains in Euclidean spaces undertaken for the development of a new approach to the classical problem about the unique determination of bounded closed convex surfaces.
We prove a complete characterization of a plane domain $U$ with smooth boundary (i.e., the Euclidean boundary fr$U$ of $U$ is a one-dimensional manifold of class $C^1$ without boundary) that is uniquely determined in the class of domains in $\mathbb{R}^2$ with smooth boundary by the condition of the local isometry of the boundaries in the relative metrics. In the case where $U$ is bounded, a necessary and sufficient condition for the unique determination of the type under consideration in the class of all bounded plane domains with smooth boundary is the convexity of $U$. If $U$ is unbounded then its unique determination in the class of all plane domains with smooth boundary by the condition of the local isometry of the boundaries in the relative metrics is equivalent to its strict convexity.