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The Sylvester-Gallai theorem asserts that given a finite number of points in the euclidean plane, either:

  1. the points are collinear
  2. there exists an ordinary line (i.e. a line that contains exactly two of the given points).

Question: Is it possible to extend this result to more general (complete) 2-manifolds (where "lines" are replaced by geodesics)? And if so, what conditions must these 2-manifolds satisfy?

I'm quite surprised that I haven't found anything about this question on the internet. It seems to be a quite natural question.

References are also much appreciated.

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I have a feeling that the correct generalization consist in studying the effect of the failure's of Euclid's fifth postulate on the two on condition due.

So you should probably study the number of points where two geodesics "in general position" intersects on your 2-manifold, and then change the two according to that. For example, on $S^2$ two geodesics intersect in two different points, and in fact the theorem as stated is false, for example pick the set of points $\{ (0,0, \pm 1),(0, \pm 1, 0), (\pm 1, 0,0) \}$ with the usual embedding of $S^2$ in $\mathbb R^3$. It is clear that every geodesic passing for two of these points meets two more points of the set, even though the 6 points are not collinear.

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  • $\begingroup$ So you really want your theorem to work with two points. Then I have no clue which condition you could impose on the curvature, perhaps non-negativeness? The theorem seems to work for hyperbolic plane by the way. $\endgroup$ – Niccolò Sep 20 '12 at 21:42
  • $\begingroup$ My first guess would be: non-positive curvature. I too think that the theorem works for hyperbolic space, did you find a proof or a reference by any chance? $\endgroup$ – Dave Sep 20 '12 at 21:58
  • $\begingroup$ No, I just thought at the Poincare's disk model. I think the theorem is equivalent to say two geodesics intersect in at most one point, and this happens in the hyperbolic space. $\endgroup$ – Niccolò Sep 20 '12 at 22:11
  • $\begingroup$ Dave, in the Klein model of hyperbolic space, geodesics are chords of the Euclidean sphere, so the Euclidean Sylvester-Gallai theorem implies the hyperbolic one. $\endgroup$ – Konrad Swanepoel Sep 25 '12 at 11:42
  • $\begingroup$ Another example: in the model of a flat torus where opposite sides of a square are identified, it is possible to embed the affine plane over any field of prime order p: just draw a square grid of side length 1/p. In these finite planes, any line passes through p points. $\endgroup$ – Konrad Swanepoel Sep 25 '12 at 11:46

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