Local rigidity of convex hypersurfaces in spaces of constant curvature

TL;DR

This paper proves a local rigidity theorem for convex hypersurfaces in spaces of constant curvature, showing that isometric convex hypersurfaces are locally congruent around points of strict convexity, extending previous smoothness assumptions.

Contribution

It extends existing rigidity results by removing the $C^1$-smoothness condition, establishing local congruence of convex hypersurfaces in constant curvature spaces.

Findings

Convex isometric hypersurfaces are locally congruent around strict convexity points.

The result applies to hypersurfaces in spaces of dimension $n \,\geq 4$.

Generalizes previous rigidity theorems by relaxing smoothness conditions.

Abstract

In this paper, we prove a local rigidity of convex hypersurfaces in the spaces of constant curvature of dimension $n\ge4$. Namely, we show that two convex isometric hypersurfaces are congruent locally around their corresponding under the isometry points of strict convexity. This result extends the result of E.P. Senkin, who showed such rigidity under the additional assumption of $C^1$-smoothness of the hypersurfaces.