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

TL;DR

The work generalizes Hilbert's non-immersion theorem to higher-dimensional hyperbolic spaces by harnessing a generalized Gauss-Bonnet framework for Riemannian polyhedra. It constructs asymptotic polyhedra from a principal adapted frame, then applies Allendoerfer–Weil integration over fibers and a symmetry-based averaging of metric connections to produce higher-dimensional Hazzidakis formulas, yielding volume bounds that contradict infinite ${\mathbb H}^{n}$ volume when immersed in ${\mathbb E}^{2n-1}$. The method blends moving frames, dual CW decompositions, and Thom form techniques to relate curvature to topology, ultimately ruling out global isometric immersions for all $n\ge 2$. This provides a comprehensive higher-dimensional obstruction grounded in differential geometry and topology with potential links to Cartan–Kähler local theory and global geometric analysis.

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 $2n-1$.