A Survey on Convex Hypersurfaces of Riemannian Manifolds

TL;DR

This survey reviews key extensions of classical rigidity theorems for convex surfaces, focusing on their generalizations to convex hypersurfaces within Riemannian manifolds, including recent results on homogeneous 3-manifolds.

Contribution

It compiles and discusses recent advances in the rigidity theory of convex hypersurfaces in Riemannian manifolds, highlighting new theorems and their implications.

Findings

Extension of classical rigidity theorems to Riemannian contexts

Inclusion of recent results on homogeneous 3-manifolds

Comprehensive overview of convex hypersurface rigidity

Abstract

We survey the main extensions of the classical Hadamard, Liebmann and Cohn-Vossen rigidity theorems on convex surfaces of $3$-Euclidean space to the context of convex hypersurfaces of Riemannian manifolds. The results we present include the one by Professor Renato Tribuzy (in collaboration with H. Rosenberg) on rigidity of convex surfaces of homogeneous $3$-manifolds.