Hilbert's Theorem, via moving frames

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).