Two classes of manifolds whose geodesic flows are integrable are defined, and their global structures are investigated. They are called Liouville manifolds and Kähler-Liouville manifolds respectively. In each case, the author finds several invariants with which they are partly classified. The classification indicates, in particular, that these classes contain many new examples of manifolds with integrable geodesic flow.

Readership

Graduate students and research mathematicians interested in differential geometry and hamiltonian mechanics.

Table of Contents

• Introduction
• Preliminary remarks and notations
• Local structure of proper Liouville manifolds
• Global structure of proper Liouville manifolds
• Proper Liouville manifolds of rank one
• Appendix. Simply connected manifolds of constant curvature

• Introduction
• Preliminary remarks and notations
• Local calculus on \(M^1\)
• Summing up the local data
• Structure of \(M-M^1\)
• Torus action and the invariant hypersurfaces
• Properties as a toric variety
• Bundle structure associated with a subset of \(\mathcal A\)
• The case where \(\#\mathcal A=1\)
• Existence theorem