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From the Wikipedia article on geodesics:

In Riemannian geometry geodesics are not the same as “shortest curves” between two points, though the two concepts are closely related. The difference is that geodesics are only locally the shortest distance between points [...]. Going the “long way round” on a great circle between two points on a sphere is a geodesic but not the shortest path between the points.

My question is:

Are there (natural, Riemannian?) geometries with more than two geodesics between two given points?

[Find a follow-up question here.]

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The wikipedia entry you cite gives the example of spheres. I don't understand what you are asking for, you already have an example. Do you want more? Any compact riemannian manifold will have many pairs of distinct points joined by several geodesics I believe. One way to see this, is to follow any geodesic farther than the radius of the manifold, then there'll be a second geodesic emanating from that point that intersects the first. – Olivier Bégassat Dec 5 '12 at 23:40
@Olivier: That's why I ask: on a sphere I can see - easily - two geodesics between two distinct points (along the great circle) but not three. – Hans Stricker Dec 5 '12 at 23:46
Take the north and south pole, there are more than two geodesics joining them, there are infinitely many. – Olivier Bégassat Dec 5 '12 at 23:48
@Olivier: you are - of course - absolutely right, and I didn't think of these exceptional cases. Can I make sense of my question by ignoring these exceptional cases: Are there geometries with more than 2 but less than $\infty$ geodesics between two given points? – Hans Stricker Dec 5 '12 at 23:53
@HansStricker: Yes, you can have any number. Just make a sphere with $n$ equidistant bulging lumps at the equator. A related thing you can ask for is the dimension of the space of such geodesics (considered as a subspace of the tangent space at the initial point) -- and this gets right to the heart of Morse theory. – Aaron Mazel-Gee Dec 5 '12 at 23:58

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