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



Reference for shortest educational path to (Riemannian) hyperbolic plane
Try sections 1-15 of this paper:


*Cannon, James W.; Floyd, William J.; Kenyon, Richard; Parry, Walter R.*, [**Hyperbolic geometry**](http://www.msri.org/communications/books/Book31/), Levy, Silvio (ed.), Flavors of geometry. Cambridge: Cambridge University Press. Math. Sci. Res. Inst. Publ. 31, 59-115 (1997). [ZBL0899.51012](https://zbmath.org/?q=an:0899.51012).


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.


