Submanifolds of Constant Negative Curvature: A Generalization of Hilbert's Theorem

TL;DR

This paper proves that hyperbolic space of any dimension cannot be isometrically embedded into Euclidean space of dimension 2n-1, using a generalized Gauss-Bonnet formula for Riemannian polyhedra.

Contribution

It extends classical results by applying a generalized Gauss-Bonnet formula to demonstrate non-embeddability of hyperbolic spaces.

Findings

Hyperbolic space of dimension n cannot be isometrically immersed into Euclidean space of dimension 2n-1.

Utilizes the generalized Gauss-Bonnet formula for Riemannian polyhedra.

Provides a new geometric obstruction based on curvature properties.

Abstract

We use the generalized Gauss-Bonnet formula for Riemannian polyhedra discovered by Allendoerfer, Weil and Chern to show that hyperbolic space of dimension $n$ has no isometric immersion into Euclidean space of dimension $2n-1$.