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