The Three Gap Theorem and Riemannian geometry
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The classical Three Gap Theorem asserts that for a natural number n and a real number p, there are at most three distinct distances between consecutive elements in the subset of [0,1) consisting of the reductions modulo 1 of the first n multiples of p. Regarding it as a statement about rotations of the circle, we find results in a similar spirit pertaining to isometries of compact Riemannian manifolds and the distribution of points along their geodesics.
KeywordsThree Gap Theorem Geodesic flow Riemannian manifold
Mathematics Subject Classification (2000)53
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- 1.Besse, A.: Manifolds All of Whose Geodesics are Closed. A Series of Modern Surveys in Mathematics, vol. 93. Springer-Verlag (1978)Google Scholar
- 4.Contreras, G.: Geodesic flows with positive topological entropy, twist maps and hyperbolicity. Ann. Math. (to appear)Google Scholar
- 5.do Carmo, M.: Riemannian Geometry. Birkhauser (1992)Google Scholar
- 7.Gaĭdukov, E.V.: Asymptotic geodesics on a Riemannian manifold nonhomeomorphic to the sphere. Soviet Math. Dokl. 7(4) (1966)Google Scholar
- 9.Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Graduate Studies in Mathematics, vol. 34. American Mathematical Society, Providence (2001)Google Scholar
- 11.Pries, C.: If all geodesics are closed on the projective plane. Archive preprint 0710.0951v1[math.DG], October (2007)Google Scholar
- 13.Świerczkowski S.: On successive settings of an arc on the circumference of a circle. Fund. Math. 46, 187–189 (1958)Google Scholar