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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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