Geodesic
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. 
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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/

[1] A. P. Kopylov, “A rigidity condition for the boundary of a submanifold in a Riemannian manifold”, Doklady Mathematics, 77:3 (2008), 340–341 | DOI | MR | Zbl

[2] A. P. Kopylov, “Unique determination of domains”, Differential Geometry and its Applications, World Sci. Publ., Hackensack, NJ, 2008, 157–169 | DOI | MR | Zbl

[3] A. V. Pogorelov, Extrinsic Geometry of Convex Surfaces, AMS, Providence, 1973 | MR | Zbl

[4] E. P. Sen'kin, “Non-flexibility of convex surfaces”, Ukr. Geom. Sb., 12 (1972), 131–152 | MR | Zbl

[5] A. P. Kopylov, “Boundary values of mappings close to isometric mappings”, Siberian Math. J., 25:3 (1985), 438–447 | DOI | MR

[6] V. A. Aleksandrov, “Isometry of domains in $\mathbb R^n$ and relative isometry of their boundaries”, Siberian Math. J., 25:3 (1985), 339–347 | DOI | MR

[7] V. A. Aleksandrov, “Isometry of domains in $\mathbb R^n$ and relative isometry of their boundaries. II”, Siberian Math. J., 26:6 (1986), 783–787 | DOI | MR

[8] M. V. Korobkov, “Necessary and sufficient conditions for unique determination of plane domains”, Siberian Math. J., 49:3 (2008 436–451) | DOI | MR

[9] M. V. Korobkov, Some rigidity theorems in Analysis and Geometry, Dis. Dokt. Fiz.-Mat. Nauk, Novosibirsk, 2008 (in Russian)

[10] M. V. Korobkov, “A criterion for the unique determination of domains in Euclidean spaces by the metrics of their boundaries induced by the intrinsic metrics of the domains”, Siberian Advances in Mathematics, 20:4 (2010), 256–284 | DOI | MR

[11] A. D. Aleksandrov, Intrinsic Geometry of Convex Surfaces, English translation, Chapman Hall/CRC Taylor Francis Group, Boca Raton, 2006 | MR