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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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    @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.2012-12-05

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