  Developing A.D. Aleksandrov's ideas, the first-named author of this article
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
2-dimensional compact connected $C^0$-submanifold with nonempty boundary in a
2-dimensional smooth connected Riemannian manifold $(X,g)$ without boundary
satisfying the condition $$\rho_{Y_1}(x,y) = \liminf_{x' \to x, y' \to y, x',y'
\in \mathop{\rm Int} Y_1} \{[l(\gamma_{x',y',\mathop{\rm Int} Y_1})]\} <
\infty,$$ if $x,y \in Y_1$. Here $\inf[l(\gamma_{x',y',\mathop{\rm Int} Y_1})]$
is the infimum of the length of smooth paths joining $x'$ and $y'$ in the
interior $\mathop{\rm Int} Y_1$ of $Y_1$. In the present paper, we first
establish that $\rho_{Y_1}$ is a metric on $Y_1$. Suppose further that $Y_1$ is
strictly convex in the metric $\rho_{Y_1}$. Consider another 2-dimensional
compact connected $C^0$-submanifold $Y_2$ of $X$ with boundary satisfying the
condition $\rho_{Y_2}(x,y) < \infty$, $x,y \in Y_2$, and assume that $\partial
Y_1$ and $\partial Y_2$ are isometric in the metrics $\rho_{Y_j}$, $j = 1,2$.
There appears the following natural question: Under which additional conditions
are the boundaries $\partial Y_1$ and $\partial Y_2$ of $Y_1$ and $Y_2$
isometric in the metric $\rho_X$ of the ambient manifold $X$? The paper is
devoted to the detailed discussions of this question. In it, we in particular
obtain a number of new results concerning the rigidity problems for the
boundaries of $C^0$-submanifolds in a Riemannian manifold. The case of $\dim
Y_j = \dim X = n$, $n > 2$, is also considered.
