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 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 Int Y1 of Y1, and extend it by continuity (operation lim ) to the boundary points of ∂Y1. In this paper the rigidity conditions are studied, i.e., when the constructed limiting metric defines ∂Y1 up to isometry of ambient space (X, g). We also consider the case dim Yj = dim X = n, n > 2.

• Kopylov A.P.
• Korobkov M.V.

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