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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

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    $\begingroup$ The half-plane model ($ds^2=(dx^2+dy^2)/y^2$) and the disc model ($ds^2=(dx^2+dy^2)/(1-x^2-y^2)^2$) both use manifolds which need only one coordinate chart. Can you ask the students to skip the charts and focus on the coordinates in this case? $\endgroup$
    – user44143
    Commented Jan 7, 2022 at 19:02
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    $\begingroup$ Check my book anton-petrunin.github.io/birkhoff it might work for you. $\endgroup$
    – Anton Petrunin
    Commented Jan 7, 2022 at 19:32
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    $\begingroup$ I'm not sure what you mean by "hyperbolic geometry is consistent with Euclidean geometry". Euclidean geometry satisfies the parallel postulate, which fails in hyperbolic geometry. Presumably, you mean that hyperbolic geometry satisfies the incidence, betweenness, and congruence axioms that Euclidean geometry does. $\endgroup$
    – Robert Bryant
    Commented Jan 7, 2022 at 19:52
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    $\begingroup$ @Robert, I think what's meant is that hyperbolic geometry is equiconsistent with Euclidean geometry, in the sense that any contradiction in either one would imply the existence of a contradiction in the other. $\endgroup$
    – Gerry Myerson
    Commented Jan 7, 2022 at 21:05
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    $\begingroup$ I think the best way to do this is to introduce Minkowski space, show what it means for a surface to be space-like, show that you can still define the first and second fundamental forms the same way you do for a surface in Euclidean space, and show that the sphere of radius $-1$ has Gauss curvature $-1$. You can use stereographic projection to get the disk and half-space models and derive all the synthetic geometric properties. This is all beautifully explained in the paper cited by @MoisheKohan, which is also available here: math.ucdavis.edu/~kapovich/RFG/cannon.pdf $\endgroup$
    – Deane Yang
    Commented Jan 8, 2022 at 23:26

4 Answers 4

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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.

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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.

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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.

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    $\begingroup$ These are all good references. The trouble is that a typical undergraduate student will need a lot of help separating the material relevant for the project from the wealth of other results/definitions/constructions contained in these books (which are primarily geared towards discrete groups and hyperbolic manifolds). $\endgroup$
    – Moishe Kohan
    Commented Jan 7, 2022 at 22:39
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I also suggest looking at Chapter 11 of Andrew Pressley's book, Elementary Differential Geometry.

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