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Motivation

I learned from Emil Artin's book Geometric Algebra that the standard incidence axioms of affine geometry (two points determine a unique line, parallel postulate, no three collinear points exist) together with (specializations of) Desargues' theorem are sufficient to translate theories of affine geometries into theories of modules over division rings.

Artin carries out explicitly the attractive construction of defining an appropriate notion of translation (a map $\tau$ with no fixed points such that the line $PQ$ determined b $P$ and $Q$ is parallel to the line$\tau(P)\tau(Q)$ determined by $\tau(P)$ and $\tau(Q)$), showing that the translations form an abelian group, and then showing that the endomorphisms of this group that fix the directions of the translations (i.e. that fix the pencil of parallel lines determined by any/each of the lines $P\tau(P)$) actually form a division ring. Intuitively, the elements of this division ring are the scalars by which translations are scaled. Then the abelian group of translations turns out to be a 2-dimensional module, and since part of Desargues' theorem states that there exist translations between any two points, this successfully coordinatizes the affine geometry.

If I have understood correctly, adding an ordering to the geometry translates into an ordering on the underlying division ring (hence making it a subfield of $\mathbb R$); distance translates into a norm on the module; angles translate into an inner product, so the above successfully models Euclidean geometry (keeping in mind the choice of the three non-collinear points: $(0,0)$, $(1,0)$, $(0,1)$) as $\mathbb R^2$.


The question

My understanding of why what Artin does actually works is that the properties of translations crucially depend on the "flatness" of affine space, which is encoded by the parallel postulate (the parallel postulate seems to allow a kind of transport of incidence data around one point to another, it seems, and Desargues's theorem as used by Artin guarantees that all such transports are coherent with one another).

Clearly, this is not true with hyperbolic geometry as it negates the parallel postulate. Nevertheless, I've read on numerous wikipedia pages that this negation is the only difference between the standard axioms of (plane) Euclidean geometry and (plane) Hyperbolic geometry. What I am curious about is whether it is possible to determine from primitive synthetic notions (points, lines, betweenness, congruence) the fact that hyperbolic geometry is a 2-dimensional Riemann surface manifold with constant negative curvature using a construction similar to the one I've read in Artin's book. Obviously, such a construction will be more sophisticated, as it would have to essentially select a coordinate chart around every point, together with transition maps (Artin's construction seems to construct one global chart for the whole of affine space).

Any references, explanations, or corrections would be much appreciated. I should perhaps mention that my nebulous endgoal is to understand hyperbolic distance, but I believe this approach, if feasible, ought to shed much intuition about (plane) hyperbolic geometry.

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    For the point "constant negative curvature", would it suffice to observe that the area of a polygon is equal to its angle defect in radians? (i.e. a triangle with an angle sum of $\pi - 1$ radians has area $1$)2011-09-04
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    @Hurkyl: yes. That's just Gauss-Bonnet theorem.2011-09-04
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    @Willie: Right, and if that's demonstration enough for Vladimir, it should be much easier to find a satisfactory reference.2011-09-05
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    @Vladimir: Also, unless I'm mistaken, it's not hard to directly prove the hyperbolic plane homeomorphic to $\mathbb{R}^2$; e.g. pick an origin and a line through it, and map the point $(x,y)$ to the point constructed by stepping $x$ units along the chosen line, erect a perpendicular at that point, then step y units along that.2011-09-05
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    @Hurkyl: ... which is a special case of the [Cartan-Hadamard theorem](http://en.wikipedia.org/wiki/Cartan-Hadamard_theorem). I must admit I don't quite understand Vladimir's original question: the motivation speaks about a connection between algebra (modules over division rings) and geometry. The question asks however about Riemann surfaces, which are objects intrinsic to (conformal/complex/algebraic/differential/riemannian; pick one) geometry. So I am not entire sure what the goal to this question is. Can the Op clarify?2011-09-05
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    Oops: where I wrote 2-dimensional "Riemann surface" I meant 2-dimensional "Riemann manifold". @Willie, as I understand it, my motivation establishes a clean correspondence between (a certain) set of axioms for "synthetic" affine geometry, and the algebraic model of affine geometry. This algebraic model, in the case of Euclidean geometry, has the extra structure that $\mathbb R^2$ has as a Hilbert space, and hence allows to interpret Euclidean geometry as an object of Riemannian geometry (I'm thinking of $\mathbb R^2$ as a global chart)...2011-09-05
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    The fact that Hyperbolic geometry differs from Euclidean geometry via the negation of a single axiom, which breaks the algebraic construction, but is nevertheless also an object of Riemannian geometry, makes me wonder about whether there is a similar correspondence between the "synthetic" axioms of the Hyperbolic plane and constant negative curvature 2d Riemann manifolds. I ask for a reference because I know barely any Riemannian geometry.2011-09-05

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