This article is cited in 1 scientific paper (total in 1 paper)
Rigidity conditions for the boundaries of submanifolds in a Riemannian manifold
Anatoly P. Kopylovab, Mikhail V. Korobkovba
a Sobolev Institute of Mathematics SB RAS,
4 Acad. Koptyug avenue, Novosibirsk, 630090, Russia
b Novosibirsk State University, Pirogova, 2, Novosibirsk, 630090,
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 $\mathopInt Y_1$ 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 $\dim Y_j = \dim X = n$, $n>2$.
Riemannian manifold, intrinsic metric, induced boundary metric, strict convexity of submanifold, geodesics, rigidity conditions.
|Russian Foundation for Basic Research
|The authors were partially supported by the RFBR for, grants 14-01-00768-a and 15-01-08275-a.
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Received in revised form: 28.04.2016
Anatoly P. Kopylov, Mikhail V. Korobkov, “Rigidity conditions for the boundaries of submanifolds in a Riemannian manifold”, J. Sib. Fed. Univ. Math. Phys., 9:3 (2016), 320–331
Citation in format AMSBIB
\by Anatoly~P.~Kopylov, Mikhail~V.~Korobkov
\paper Rigidity conditions for the boundaries of submanifolds in a Riemannian manifold
\jour J. Sib. Fed. Univ. Math. Phys.
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A. P. Kopylov, “On the unique determination of domains by the condition of the local isometry of the boundaries in the relative metrics”, Sib. elektron. matem. izv., 14 (2017), 59–72
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