I am teaching an undergraduate class for math majors on axiomatic geometry, culminating in the proof that hyperbolic geometry is equiconsistent* with Euclidean geometry. I would like to make an end-of-term project for them to write about an alternate route to the hyperbolic plane via Riemannian geometry, but every resource I know spends time on atlases before turning to the metric.
Does anyone know of a reference that deals with the metric first, so that we can go directly from calculus to the hyperbolic plane (without having to deal with atlases)?
*thanks for the correction, Robert Bryant & Gerry Myerson
Best Answer
Try sections 1-15 of this paper:
Cannon, James W.; Floyd, William J.; Kenyon, Richard; Parry, Walter R., Hyperbolic geometry, Levy, Silvio (ed.), Flavors of geometry. Cambridge: Cambridge University Press. Math. Sci. Res. Inst. Publ. 31, 59-115 (1997). ZBL0899.51012.
It introduces the bare minimum of Riemannian geometry needed for the task, namely for domains in ${\mathbb R}^n$. Geodesics are identified with circular arcs not using the connection and geodesic equation (these are never even mentioned in the paper) but using certain retractions. Pretty much everything is written on the vector-calculus level, so undergraduate students can handle this.