TITLE: Cohn-Vossen rigidity theorem in hyperbolic space
QUESTION [5 upvotes]: 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

REPLY [2 votes]: 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).