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.

  • $\begingroup$ 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 $1$) $\endgroup$ – Hurkyl Sep 4 '11 at 7:10
  • $\begingroup$ @Hurkyl: yes. That's just Gauss-Bonnet theorem. $\endgroup$ – Willie Wong Sep 4 '11 at 13:13
  • $\begingroup$ @Willie: Right, and if that's demonstration enough for Vladimir, it should be much easier to find a satisfactory reference. $\endgroup$ – Hurkyl Sep 5 '11 at 14:13
  • $\begingroup$ @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. $\endgroup$ – Hurkyl Sep 5 '11 at 14:15
  • $\begingroup$ @Hurkyl: ... which is a special case of the 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? $\endgroup$ – Willie Wong Sep 5 '11 at 15:34

Yes, one can develop hyperbolic geometry synthetically. One replaces the parallel postulate by the axiom that through any point $P$ disjoint from a line $\ell$, there are at least two distinct lines passing through $P$ that are disjoint from $\ell$.

Building on this, one can go on to prove that similar triangles are necessarily congruent. One also finds a canonical unit of length, which in non-synthetic terms is chosen so that the curvature is scaled to be $-1$ (rather than just some arbitrary negative number).

Unfortunately I don't know where to read this; I learnt it from a beautiful little book from my (then) local library called (if I remember correctly) Euclidean and non-Euclidean geometry many years ago, but in more recent years I've not been able to find this book, and I don't know an alternate reference (but surely there are many!).

Added: Thanks to Willie's comment below, I am now able to state the reference I learned this from: Foundations of Euclidean and non-Euclidean geometry, by E. Golos.

As a side note, rereading the relevant section, I saw that I misstated the relevant axiom in my first paragraph: the particular axiom in that text, which apparently follows a suggestion of Hilbert, is to assume that given a line $\ell$ and a point $P$ not on $\ell$, there exist two rays through $P$, neither of which intersects $\ell$, but such that any ray lying strictly between them does intersect $\ell$. (This is stronger than the axiom stated in my first paragraph; Golos's text suggests that weaker axioms are possible, but I haven't thought about what they might be, and which precise formulation is optimal.)

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    $\begingroup$ One reference I like is "Higher geometry" by Efimov, amazon.com/Higher-geometry-N-V-Efimov/dp/B0000EHA9S $\endgroup$ – Andrés E. Caicedo Sep 5 '11 at 20:04
  • $\begingroup$ @Willie: Dear Willie, No, unfortunately. The book I learnt from was not really a text-book, but more like a little monograph, with a purely axiomatic development (no formulas or anything like that; purely synthetic). I think that it was a simplified version of Hilbert's axioms (and it only treated line segments and angles, not circles). Thanks, though! Best wishes, $\endgroup$ – Matt E Sep 6 '11 at 2:00
  • $\begingroup$ Dear Matt, From your description it could have been an old edition (first or second) of Marvin Jay Greenberg's book Euclidean and Non-Euclidean Geometries: Development and History. Best wishes, $\endgroup$ – t.b. Sep 6 '11 at 11:59
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    $\begingroup$ Dear Matt, sorry to bother you again, but may it be Ellery Golos' Foundations of Euclidean and non-Euclidean geometry? Unfortunately I cannot get at a copy (the local copy is in a library which I am not registered to use) to compare against your description above, so am making this guess more based on citations to it, and on the American Mathematical Monthly review of the book $\endgroup$ – Willie Wong Sep 8 '11 at 23:41
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    $\begingroup$ @Willie: Dear Willie, I checked in the library today, and this is indeed the book I had in mind. Thank you very much! Best wishes, $\endgroup$ – Matt E Sep 10 '11 at 0:33

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