Rigidity of convex hypersurfaces, Rigidity of convex hypersurfaces in multidimensional spaces of constant curvature

TL;DR

This paper extends the rigidity theorem for convex hypersurfaces, removing smoothness assumptions, and proves similar results in spaces of constant curvature, advancing understanding of geometric uniqueness in higher dimensions.

Contribution

It removes smoothness restrictions and establishes rigidity results for convex hypersurfaces in Euclidean and constant curvature spaces.

Findings

Rigidity of convex hypersurfaces in Euclidean spaces for arbitrary compact cases.

Extension of rigidity results to spaces of constant curvature.

Removal of smoothness assumptions in the rigidity theorem.

Abstract

In 1972, E. P. Senkin generalized the celebrated theorem of A. V. Pogorelov on unique determination of compact convex surfaces by their intrinsic metrics in the Euclidean 3-space $E^3$ to higher dimensional Euclidean spaces $E^{n+1}$ under a mild assumption on the smoothness of the hypersurface. In this paper, we remove this assumption and thus establish this rigidity result for arbitrary compact, convex hypersurfaces in $E^{n+1}$, $n \ge 3$. We also prove the corresponding results in other model spaces of constant curvature.