Mathematical Research Letters

Volume 9, Issue 1, January 2002 pp. 105-115.

Completely integrable torus actions on symplectic cones

Authors: Eugene Lerman and Nadya Shirokova
Author institution: University of Illinois, Urbana

Summary: We study completely integrable torus actions on symplectic cones (equivalently, completely integrable torus actions on contact manifolds). We show that if the cone in question is the punctured cotangent bundle of a torus then the action has to be free. From this it follows easily, using hard results of Ma\~ne and of Burago and Ivanov, that a metric on a torus whose geodesic flow admits global action-angle coordinates is necessarily flat thereby proving a conjecture of Toth and Zelditch.

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