On the topology of manifolds with completely integrable geodesic flows. II
type: Item
instance of: scholarly article
title: On the topology of manifolds with completely integrable geodesic flows. II
zbMATH Open document ID: 0801.58037
DOI: 10.1016/0393-0440(94)90036-1
author: Gabriel P. Paternain
published in: Journal of Geometry and Physics
publication date: 1994-11-29T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 569291
zbMATH Keywords: entropy
zbMATH Keywords: integrable geodesic flows
zbMATH Keywords: loop space homology
reviewed by: D. J. Hurley
MaRDI profile type: Publication
full work available at URL: 0393-0440(94)90036-1
OpenAlex ID: W2007033252
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description: scientific article; zbMATH DE number 569291