Integrable geodesic flows on n-step nilmanifolds

Recent examples of Liouville-integrable geodesic flows on non-simply connec ted manifolds have shown that the topological implications of C-infinity Li ouville integrability are dramatically different from the implications of r eal-analytic integrability. In particular a geodesic flow can be both smoot hly integrable and have positive topological entropy [A.V. Bolsinov, I.A. T aimanov, Russ. Math. Surveys 54 (4) (1999) 833-835]. The examples of Bolsin ov and Taimanov, and of Butler [L. Butler, CR Math. Rep. Acad. Sci. Can. 21 (4) (1999) 127-131] are constructed from left-invariant metrics on Lie gro ups. In this paper, the degeneracy of the Poisson tensor on the dual algebr a is shown to be the source of the large number of commuting first integral s, and additional examples of integrable geodesic flows are constructed on n-step nilmanifolds. (C) 2000 Elsevier Science B.V. All rights reserved. MS C: primary 58F07; secondary 58F17, 53C22.