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There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: http://www.math.brown.edu/~deigen/chern.pdf

Any isometry between two closed smooth convex surfaces in the Euclidean space $\mathbb{R}^3$ is established by an isometry of $\mathbb{R}^3$.

Is the same result true if one considers convex surfaces with the same assumptions in the hyperbolic space $\mathbb{H}^3$ instead of $\mathbb{R}^3$?

This post is a continuation and a more precise version of Extendability of isometries of convex surfaces

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Hsiung-Liu: Generalization of the rigidity theorem of Cohn-Vossen prove the rigidity theorem for surfaces in hyperbolic 3-space, assuming that their second fundamental form is positive definite and a condition on the directions of normal vectors (these assumptions replacing the convexity assumption one needs in Euclidean 3-space).

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  • $\begingroup$ On my computer I am able to view only the first page of the paper. Thm 1.1, which is claimed to be the purpose of the paper, uses apparently some extra assumptions on surfaces, and also the conclusion does not look exactly the same as in my question. Does the rest of the paper contain an answer? $\endgroup$
    – asv
    Commented Aug 9, 2019 at 12:38
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    $\begingroup$ @MKO, The proof for Euclidean space uses an integral formula involving the support function of the convex body. In hyperbolic space it's not as clear how to define the support function of a convex body. So the proof does not transfer over cleanly. If you define a convex body in hyperbolic space as an intersection of horoballs (analogous to half spaces in Euclidean space) and use Busemann functions to define the analogue of a support function, then there is a chance that the Cohn-Vossen proof can be used in that setting. $\endgroup$
    – Deane Yang
    Commented Aug 9, 2019 at 19:03

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