On the topology of manifolds with completely integrable geodesic flows. II
The author continues his investigations, concerning topological properties of manifolds with completely integrable geodesic flows, which he began in Ergodic Theory Dyn. Syst. 12, No. 1, 109-122 (1993; Zbl 0735.58028). It is shown that if \(M\) is a compact simply connected Riemannian manifold whose geodesic flow is completely integrable with non-degenerate first integrals, then \(\pi_ 1(M)\) has sub-exponential growth. If \(\pi_ 1(M)\) is finite, then the loop space homology of \(M\), \(\sum^ k_{i=1} \dim H_ i(\Omega,M,K)\) grows sub-exponentially for any coefficient field \(K\). It is also shown that if for some point \(p\in M\), the geodesic flow of \(M\) admits action-angle coordinates with singularities in a neighbourhood of every vector in the unit sphere at \(p\), then \(M\) is \(Z\)-elliptic, i.e. \(\sum^ k_{i=1} \dim H_ i(\Omega,M,K)\) grows polynomially for any coefficient field \(K\) [\textit{Y. Félix}, \textit{S. Halperin} and \textit{J.-C. Thomas}, Bull. Am. Math. Soc., New Ser. 25, No. 1, 69-73 (1991; Zbl 0726.55006)].
scientific article; zbMATH DE number 569291
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