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 J. Sib. Fed. Univ. Math. Phys., 2016, Volume 9, Issue 3, Pages 320–331 (Mi jsfu490)

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, Russia

Abstract: Developing A.D. Aleksandrov's ideas, the first author proposed the following approach to study of rigidity problems for the boundary of a $C^0$-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 $\dim Y_j = \dim X = n$, $n>2$.

Keywords: Riemannian manifold, intrinsic metric, induced boundary metric, strict convexity of submanifold, geodesics, rigidity conditions.

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-00768_a15-01-08275_a The authors were partially supported by the RFBR for, grants 14-01-00768-a and 15-01-08275-a.

* Author to whom correspondence should be addressed

DOI: https://doi.org/10.17516/1997-1397-2016-9-3-320-331

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UDC: 517.95
Accepted: 26.05.2016
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Citation: 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
\Bibitem{KopKor16} \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. \yr 2016 \vol 9 \issue 3 \pages 320--331 \mathnet{http://mi.mathnet.ru/jsfu490} \crossref{https://doi.org/10.17516/1997-1397-2016-9-3-320-331} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000412010000007} 

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This publication is cited in the following articles:
1. 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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