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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 $x,y \\in Y_1$. 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