Rigidity Conditions for the Boundaries of Submanifolds in a Riemannian Manifold

Developing A.D. Aleksandrov's ideas, the first author proposed the following approach to study of rigidity problems for the boundary of a C^sup 0^-submanifold in a smooth Riemannian manifold. Let Y^sub 1^ be a two-dimensional compact connected C^sup 0^-submanifold with non-empty boundary in some smooth two-dimensional 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 Int Y^sub 1^ of Y^sub 1^, and extend it by continuity (operation lim) to the boundary points of ∂Y^sub 1^. In this paper the rigidity conditions are studied, i.e., when the constructed limiting metric defines ∂Y^sub 1^ up to isometry of ambient space (X, g). We also consider the case dim Y^sub j^ = dim X = n, n > 2.