Integrable Geodesic Flows on Two-Dimensional Surfaces

Geodesic flows of Riemannian metrics on manifolds are one of the classical objects in geometry. A particular place among them is occupied by integrable geodesic flows. We consider them in the context of the general theory of integrable Hamiltonian systems, and in particular, from the viewpoint of a new topological classification theory, which was recently developed for integrable Hamiltonian systems with two degrees of freedom. As a result, we will see that such a new approach is very useful for a deeper understanding of the topology and geometry of integrable geodesic flows. The main object to be studied in our paper is the class of integrable geodesic flows on two-dimensional surfaces. There are many such flows on surfaces of small genus, in particular, on the sphere and torus. On the contrary, on surfaces of genus 9 > 1, no such flows exist in the analytical case. One of the most important and interesting problems consists in the classification of integrable flows up to different equivalence relations such as (1) an isometry, (2) the Liouville equivalence, (3) the trajectory equivalence (smooth and continuous), and (4) the geodesic equivalence. In recent years, a new technique was developed, which gives, in particular, a possibility to classify integrable geodesic flows up to these kinds of equivalences. This technique is presented in our paper, together with various applications. The first part of our book, namely, Chaps.

Inhalt

1. Basic Notions.- 1.1. Linear Symplectic Geometry.- 1.2. Symplectic and Poisson Manifolds.- 1.3. Local Properties of Symplectic Manifolds.- 1.4. Liouville Integrable Hamiltonian Systems. Liouville Theorem.- 1.5. Nonresonant and Resonant Systems.- 1.6. Rotation Number.- 1.7. Momentum Mapping of an Integrable Hamiltonian System and Its Bifurcation Diagram.- 1.8. Nondegenerate Singularities of the Momentum Mapping and Bott Functions.- 1.9. Bott Integrals from the Point of View of the Four-Dimensional Symplectic Manifold.- 1.10. Main Types of Equivalence of Dynamical Systems.- 2. Topology of Foliations Generated by Morse Functions on Two-Dimensional Surfaces.- 2.1. Simple Morse Functions.- 2.2. Reeb Graph of a Morse Function.- 2.3. Concept of an Atom.- 2.4. Simple Molecules.- 2.5. Complicated Atoms.- 2.6. Classification of Atoms.- 2.7. Notion of a Molecule.- 2.8. Approximation of Complicated Molecules by Simple Ones.- 3. Rough Liouville Equivalence of Integrable Systems with Two Degrees of Freedom.- 3.1. Classification of Nondegenerate Critical Submanifolds on Isoenergy 3-Surfaces.- 3.2. The Topological Structure of a Neighborhood of a Singular Leaf.- 3.3. Topologically Stable Hamiltonian Systems.- 3.4. 2-Atoms and 3-Atoms.- 3.5. Classification of 3-Atoms.- 3.6. 3-Atoms as Bifurcations of Liouville Tori.- 3.7. The Molecule of an Integrable System.- 4. Liouville Equivalence of Iintegrable Systems with Two Degrees of Freedom.- 4.1. Admissible Coordinate Systems on the Boundary of a 3-Atom.- 4.2. Gluing Matrices and Superfluous Frames.- 4.3. Invariants (Numerical Marks) r, ?, and n.- 4.4. The Marked Molecule.- 4.5. Influence of the Orientation.- 4.6. Realization Theorem.- 4.7. Simple Examples of Molecules.- 4.8. Hamiltonian Systems with Critical Klein Bottles.- 5. TrajectoryClassification of Integrable Systems with Two Degrees of Freedom.- 5.1. Rotation Function and Rotation Vector.- 5.2. Reduction of the Three-Dimensional.- 5.3. General Concept of Constructing Trajectory Invariants of Integrable Hamiltonian Systems.- 6. Integrable Geodesic Flows on Two-Dimensional Surfaces.- 6.1. Statement of the Problem.- 6.2. Topological Obstructions to Integrability of Geodesic Flows on Two-Dimensional Surfaces.- 6.3. Two Examples of Integrable Geodesic Flows.- 6.4. Riemannian Metrics Whose Geodesic Flows are Integrable by Means of Linear or Quadratic Integrals. Local Theory.- 6.5. Linearly and Quadratically Integrable Geodesic Flows on Closed Surfaces.- 7. Liouville Classification of Integrable Geodesic Flows on Two-Dimensional Surface.- 7.1. Liouville Classification of Linearly and Quadratically Integrable Geodesic Flows on the Torus.- 7.2. Liouville Classification of Linearly and Quadratically Integrable Geodesic Flows on the Klein Bottle.- 7.3. Liouville Classification of Linearly and Quadratically Integrable Geodesic Flows on the Two-Dimensional Sphere.- 7.4. Liouville Classification of Linearly and Quadratically Integrable Geodesic Flows on the Projective Plane.- 8. Trajectory Classification of Integrable Geodesic Flows on Two-Dimensional Surfaces.- 8.1. Case of the Torus.- 8.2. Case of the Sphere.- 8.3. Examples of Integrable Geodesic Flows on the Sphere.- 8.4. Non-Triviality of Trajectory Equivalence Classes and Metrics with Closed Geodesics.- 9. Maupertuis Principle and Geodesic Equivalence.- 9.1. General Maupertuis Principle.- 9.2. Maupertuis Principle in Rigid Body Dynamics.- 9.3. Maupertuis Principle and an Explicit Form of the Metric on the Sphere, Generated by a Quadratic Hamiltonian on the Lie Algebra e(3).- 9.4. Classical Cases of Integrability in Rigid Body Dynamics and Related Integrable Geodesic Flows on the Sphere.- 9.5. Conjecture on Geodesic Flows with Integrals of High Degree.- 9.6. Dini Theorem and the Geodesic Equivalence of Riemannian Metrics.- 9.7. Generalized Dini-Maupertuis Principle.- 9.8. Trajectory Equivalence of the Neumann Problem and Jacobi Problem.- 9.9. Explicit Forms of Some Remarkable Hamiltonians and Their Integrals in Separating Variables.- 10. Euler Case in Rigid Body Dynamics and Jacobi Problem About Geodesics on the Ellipsoid. Trajectory Isomorphism.- 10.1. Introduction.- 10.2. Jacobi Problem and Euler Case.- 10.3. Liouville Foliations.- 10.4. Rotation Functions.- 10.5. The Main Theorem.- 10.6. Smooth Invariants.- 10.7. Topological Non-Conjugacy of the Jacobi Problem and the Euler Case.