Rigidity conditions for the boundaries of submanifolds in a Riemannian manifold
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 9 (2016) no. 3, pp. 320-331
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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 $\mathop{\rm Int} Y_1$ 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.
@article{JSFU_2016_9_3_a6,
author = {Anatoly P. Kopylov and Mikhail V. Korobkov},
title = {Rigidity conditions for the boundaries of submanifolds in a {Riemannian} manifold},
journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
pages = {320--331},
publisher = {mathdoc},
volume = {9},
number = {3},
year = {2016},
language = {en},
url = {http://geodesic.mathdoc.fr/item/JSFU_2016_9_3_a6/}
}
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%0 Journal Article %A Anatoly P. Kopylov %A Mikhail V. Korobkov %T Rigidity conditions for the boundaries of submanifolds in a Riemannian manifold %J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika %D 2016 %P 320-331 %V 9 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/item/JSFU_2016_9_3_a6/ %G en %F JSFU_2016_9_3_a6
Anatoly P. Kopylov; Mikhail V. Korobkov. Rigidity conditions for the boundaries of submanifolds in a Riemannian manifold. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 9 (2016) no. 3, pp. 320-331. http://geodesic.mathdoc.fr/item/JSFU_2016_9_3_a6/