Hilbert's Theorem, via moving frames

TL;DR

This paper provides a simplified proof that the hyperbolic plane cannot be smoothly embedded in three-dimensional Euclidean space, using moving frames and connection forms to connect geometric properties with the impossibility.

Contribution

It introduces a novel proof technique employing moving frames and connection forms to demonstrate the non-embeddability of the hyperbolic plane in Euclidean 3-space.

Findings

Hyperbolic plane cannot be isometrically immersed in Euclidean 3-space.

Uses moving frames to simplify the proof of non-embeddability.

Connects geometric invariants with the impossibility of embedding.

Abstract

We present a proof that the hyperbolic plane cannot be isometrically immersed in Euclidean $3$-space by a $C^\infty$ map. Ideas from many topics in (essentially) undergraduate mathematics are applied; the use of moving frames and connection forms to express the geometry simplifies the outline of the proof, compared to, say, using coordinate patches and Christoffel symbols. The key transition is from expressions in terms of the principal directions on the immersed surface (which give access to the Gaussian curvature) to expressions in terms of the asymptotic directions (which yield a coordinate system and give access to surface area).