# Extension of the Sylvester-Gallai theorem?

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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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.