Reference for shortest educational path to (Riemannian) hyperbolic plane 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
 A: 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.
A: Ted Shifrin's course notes on curves and surfaces has a nice introduction to hyperbolic geometry in the plane. You have to refer to the earlier part of the book for the notation related to Riemannian metrics. (That discussion is local, and the hyperbolic plane has only one chart, so there's no need for general discussion of charts.) There are some exercises about parallelism, the isometry group, and a couple of the models of the hyperbolic plane.
A: These books may fit your need.
Svetlana Katok, Fuchsian groups
Alan Beardon, The geometry of discrete groups, ch.7
The book of Ratcliffe, Foundations of hyperbolic manifolds, is slightly more advanced and deals with higher dimensional hyperbolic spaces.
A: I also suggest looking at Chapter 11 of Andrew Pressley's book, Elementary Differential Geometry.
