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