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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 rigidity problems of domains in Euclidean spaces carried out for developing a new approach to the classical problem of the unique determination of bounded closed convex surfaces [A.V.Pogorelov, Extrinsic Geometry of Convex Surfaces, AMS, Providence (1973)], rather completely presented in [A.P.Kopylov, On the unique determination of domains in Euclidean spaces, J. of Math. Sciences, 153, no.6, 869-898 (2008)]. We give a complete characterization of a plane domain $U$ with smooth boundary (i.e., the Euclidean boundary $\mathop{\rm fr}U$ 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 boundaries by the condition of the local isometry of the boundaries in the relative metrics. If $U$ is bounded then the convexity of $U$ is a necessary and sufficient condition for the unique determination of this kind in the class of all bounded plane domains with smooth boundaries. If $U$ is unbounded then its unique determination in the class of all plane domains with smooth boundaries by the condition of the local isometry of the boundaries in the relative metrics is equivalent to its strict convexity. In the last section, we consider the case of space domains. We prove a theorem on the unique determination of a strictly convex domain in $\mathbb R^n$, where $n \ge 2$, in the class of all $n$-dimensional domains by the condition of the local isometry of the Hausdorff boundaries in the relative metrics, which is a generalization of A.D.Aleksandrov's theorem on the unique determination of a strictly convex domain by the condition of the (global) isometry of the boundaries in the relative metrics.