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 $C^0$-submanifold in a smooth Riemannian manifold. Let $Y_1$ be a two-dimensional compact connected $C^0$-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 $\mathop{\rm Int} Y_1$ of $Y_1$, and extend it by continuity (operation $ \varliminf$) to the boundary points of $\partial Y_1$. In this paper the rigidity conditions are studied, i.e., when the constructed limiting metric defines $\partial Y_1$ up to isometry of ambient space $(X,g)$. We also consider the case $\dim Y_j = \dim X = n$, $n>2$.