Mathematics & Physics 2016, 9(3), 320–331 УДК 517.95 Rigidity Conditions for the Boundaries of Submanifolds in a Riemannian Manifold Anatoly P.Kopylov∗ Mikhail V. Korobkov† Sobolev Institute of Mathematics SB RAS 4 Acad. Koptyug avenue, Novosibirsk, 630090 Novosibirsk State University Pirogova, 2, Novosibirsk, 630090 Russia Received 20.03.2016, received in revised form 28.04.2016, accepted 26.05.2016 Developing A.D. Aleksandrov’s ideas, the first author proposed the following approach to study of rigidity problems for the boundary of a C0-submanifold in a smooth Riemannian manifold. <...> Let Y1 be a two-dimensional compact connected C0-submanifold with non-empty boundary in some smooth twodimensional Riemannian manifold (X, g) without boundary. <...> Let us consider the intrinsic metric (the infimum of the lengths of paths, connecting a pair of points".) of the interior IntY1 of Y1, and extend it by continuity (operation lim ) to the boundary points of ∂Y1. <...> Keywords: Riemannian manifold, intrinsic metric, induced boundary metric, strict convexity of submanifold, geodesics, rigidity conditions. <...> Introduction: unique determination of surfaces by their relative metrics on boundaries A classical theorem says (see [3]): If two bounded closed convex surfaces in the threedimensional Euclidean space are isometric in their intrinsic metrics then they are equal, i.e., they can be matched by a motion. <...> The problems of unique determination of closed convex surfaces by their intrinsic metrics goes back to the result of Cauchy, obtained already in 1813, that any closed convex polyhedrons P1 and P2 (in the three-dimensional Euclidean space) that are equally composed of congruent faces are equal. <...> Since then this problem has been studied by many people for about 140 years (for example, by Minkowski, Hilbert,Weyl, Blaschke, Cohn-Vossen, Aleksandrov, Pogorelov and other prominent mathematicians (see, for instance, the historical overview in [3], Chapter 3); finally, its complete solution, which is just the theorem we have cited at the beginning, was obtained by A. V.Pogorelov. <...> The following model situation illustrates the essence of this approach fairly well: ∗apkopylov@yahoo.com †korob@math.nsc.ru ⃝ Siberian Federal University. <...> All rights reserved c – 320 – Anatoly P.Kopylov, Mikhail V. Korobkov Rigidity <...>