If any two dinstict points in a complete Riemannian manfiold can only be joined by two different geodesics, is the Riemannian manifold isometric to round sphere?
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2$\begingroup$ For a sphere, two opposite points can be joined by many geodesics. I wonder if you're looking for something like the closed-loop criterion mentioned in the answer about Zoll surfaces below. $\endgroup$– Martin M. W.Commented Oct 27 at 13:19
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$\begingroup$ For the sphere you will always get more than two geodesics between some pairs of points. But, in $\mathbb{R}\mathrm{P}^2$, any two points can only be joined exactly two different (simple) geodesics. $\endgroup$– Anton PetruninCommented Oct 28 at 0:51
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$\begingroup$ Yes, Martin M.W. and Anton Petrunin are right, antipodal points on sphere can be joined by infinitely many geodesics. Basically, I wanted to find some necessary conditions on geodesics to characterize space forms. I know little about this. $\endgroup$– Y.SunCommented Oct 28 at 1:24
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1 Answer
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A Zoll surface is a 2-dimensional Riemannian manifold all of whose geodesics are closed smooth loops of equal length. There is an infinite dimensional family of Zoll surfaces; this was first proved by Otto Zoll in 1903, but there are also very different and beautiful approaches of Victor Guillemin, and of Lebrun and Mason.
Guillemin, Victor (1976), "The Radon transform on Zoll surfaces", Advances in Mathematics, 22 (1): 85–119, doi:10.1016/0001-8708(76)90139-0
LeBrun, Claude; Mason, L.J. (July 2002), "Zoll manifolds and complex surfaces", Journal of Differential Geometry, 61 (3): 453–535, arXiv:math/0211021, doi:10.4310/jdg/1090351530
