Rigidity Conditions for the Boundaries of Submanifolds in a Riemannian Manifold

Abstract

Developing A.D. Aleksandrov’s ideas, the first author proposed the following approach to study of rigidity problems for the boundary of a C-submanifold in a smooth Riemannian manifold. Let Y1 be a two-dimensional compact connected C-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 IntY1 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 dimYj = dimX = n, n > 2.