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Two Classes of Riemannian Manifolds Whose Geodesic Flows Are Integrable

Kazuyoshi Kiyohara The Mathematical Society of Japan, Tokyo, Japan
Available Formats:
Electronic ISBN: 978-1-4704-0208-2
Product Code: MEMO/130/619.E
List Price: $50.00 MAA Member Price:$45.00
AMS Member Price: $30.00 Click above image for expanded view Two Classes of Riemannian Manifolds Whose Geodesic Flows Are Integrable Kazuyoshi Kiyohara The Mathematical Society of Japan, Tokyo, Japan Available Formats:  Electronic ISBN: 978-1-4704-0208-2 Product Code: MEMO/130/619.E  List Price:$50.00 MAA Member Price: $45.00 AMS Member Price:$30.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 1301997; 143 pp
MSC: Primary 53; 58; Secondary 70;

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.

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

• Chapters
• Part 1. Liouville manifolds
• Introduction
• 1. Local structure of proper Liouville manifolds
• 2. Global structure of proper Liouville manifolds
• 3. Proper Liouville manifolds of rank one
• Appendix. Simply connected manifolds of constant curvature
• Part 2. Kähler-Liouville manifolds
• Introduction
• 1. Local calculus on $M^1$
• 2. Summing up the local data
• 3. Structure of $M-M^1$
• 4. Torus action and the invariant hypersurfaces
• 5. Properties as a toric variety
• 6. Bundle structure associated with a subset of $\mathcal {A}$
• 7. The case where $\#\mathcal {A}=1$
• 8. Existence theorem
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Volume: 1301997; 143 pp
MSC: Primary 53; 58; Secondary 70;

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.

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

• Chapters
• Part 1. Liouville manifolds
• Introduction
• 1. Local structure of proper Liouville manifolds
• 2. Global structure of proper Liouville manifolds
• 3. Proper Liouville manifolds of rank one
• Appendix. Simply connected manifolds of constant curvature
• Part 2. Kähler-Liouville manifolds
• Introduction
• 1. Local calculus on $M^1$
• 2. Summing up the local data
• 3. Structure of $M-M^1$
• 4. Torus action and the invariant hypersurfaces
• 5. Properties as a toric variety
• 6. Bundle structure associated with a subset of $\mathcal {A}$
• 7. The case where $\#\mathcal {A}=1$
• 8. Existence theorem
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