Geodesic
Topological obstructions to integrability of geodesic flows on non-simply-connected manifolds
Izvestiya. Mathematics , Tome 30 (1988) no. 2, pp. 403-409

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In this paper, (Liouville) integrability of geodesic flows on non-simply-connected manifolds is studied. In particular, the following result is obtained: A geodesic flow on a real-analytic Riemannian manifold cannot be integrable in terms of analytic functions if either 1) the fundamental group of the manifold contains no commutative subgroup of finite index, or 2) the first Betti number of the manifold over the field of rational numbers is greater than the dimension (the manifold is assumed to be closed). Bibliography: 11 titles. 
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     author = {I. A. Taimanov},
     title = {Topological obstructions to integrability of geodesic flows on non-simply-connected manifolds},
     journal = {Izvestiya. Mathematics },
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     volume = {30},
     number = {2},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1988_30_2_a12/}
}
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I. A. Taimanov. Topological obstructions to integrability of geodesic flows on non-simply-connected manifolds. Izvestiya. Mathematics , Tome 30 (1988) no. 2, pp. 403-409. http://geodesic.mathdoc.fr/item/IM2_1988_30_2_a12/