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On the topology of manifolds with completely integrable geodesic flows

Published online by Cambridge University Press:  19 September 2008

Gabriel P. Paternain
Gabriel P. Paternain
Affiliation:
Department of Mathematics, SUNY Stony Brook, Stony Brook, NY 11794, USA
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Abstract

We show that if M is a compact simply connected Riemannian manifold whose geodesic flow is completely integrable with periodic integrals, then M is rationally elliptic, i.e. the rational homotopy of M is finite dimensional. We also show that rational ellipticity is shared by simply connected compact manifolds whose cotangent bundle admits a multiplicity free compact action that leaves invariant the Hamiltonian associated with some Riemannian metric. In particular it follows that if M is a Riemannian manifold whose geodesic flow is completely integrable by the Thimm method, then M is rationally elliptic. Other questions concerning the global behaviour of geodesics on homogeneous spaces are discussed.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 1 , March 1992 , pp. 109 - 121
Copyright
Copyright © Cambridge University Press 1992

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