A famous theorem of Hilbert says that there is no smooth immersion of the hyperbolic plane in 3-dimensional Euclidean space. The expositions of this that I know of (in eg do Carmo’s book on curves/surfaces, and in Spivak vol 3) are very analytic and non-geometric, with lots of delicate formulas. However, in Thurston’s book on 3d geometry and topology, he suggests that this is really a geometric fact that should be proved by looking at the lines of curvature and how they twist around as you go off to infinity.

Question: does anyone know any alternate proof of this (or at least novel expositions of the usual ones) that emphasize the geometry and minimize the pages and pages of formulas? I don’t care about minimizing smoothness assumptions (I would rather have analytic simplicity!).