A Rigidity Theorem for Hypersurfaces in Higher Dimensional Space Forms

TL;DR

This paper generalizes the classical Cohn-Vossen theorem, establishing a rigidity result for higher-dimensional hypersurfaces in space forms, showing that isometric convex hypersurfaces are congruent.

Contribution

It extends the classical rigidity theorem from 3D surfaces to higher-dimensional hypersurfaces in space forms.

Findings

Higher-dimensional hypersurfaces in space forms exhibit rigidity similar to classical surfaces.

Isometric convex hypersurfaces in $N^{n+1}(K)$ are congruent.

The generalization applies for dimensions $n \\ge 2$.

Abstract

The classical Cohn-Vossen theorem states that two isometric compact convex surfaces in $\mathbb{R}^{3}$ are congruent. In this short note, we generalize the classical Cohn-Vossen Theorem to higher dimensional surfaces in space form $N^{n+1}(K)$ for $n\ge 2$.