Geodesic equivalence and integrability (original) (raw)

Geometrical Obstruction to the Integrability of Geodesic Flows

2002

Resorting to classical techniques of Riemannian geometry we develop a geometrical method suitable to investigate the nonintegrability of geodesic flows and of natural Hamiltonian systems. Then we apply such method to the Anisotropic Kepler Problem (AKP) and we prove that it is not analytically integrable.

Integrability Criterion of Geodesical Equivalence. Hierarchies

Acta Applicandae Mathematicae - ACTA APPL MATH, 1999

We prove that the Riemannian metrics g and \bar g$$ (given in `general position") are geodesically equivalent if and only if some canonically given functions are pairwise commuting integrals of the geodesic flow of the metric g. This theorem is a multidimensional generalization of the well-known Dini theorem proved in the two-dimensional case. A hierarchy of completely integrable Riemannian metrics is assigned to any pair of geodesically equivalent Riemannian metrics. We show that the metrics of the standard ellipsoid and the Poisson sphere lie in such an hierarchy.

On the integrability of the geodesic flow on a Friedmann–Robertson–Walker spacetime

Physica Scripta, 2018

We study the geodesic flow on the cotangent bundle T * M of a Friedmann-Robertson-Walker (FRW) spacetime (M, g). On this bundle, the Hamilton-Jacobi equation is completely separable and this property allows us to construct four linearly independent integrals in involution, i.e. Poisson commuting amongst themselves and pointwise linearly independent. As a consequence, the geodesic flow on an FRW background is completely integrable in the Liouville-Arnold sense. For a spatially flat or spatially closed universe, we construct submanifolds that remain invariant under the action of the flow. For a spatially closed universe these submanifolds are topologically R × S 1 × S 1 × S 1 , while for a spatially flat universe they are topologically R×R×S 1 ×S 1. However, due to the highly symmetrical nature of the background spacetime, the four integrals in involution also admit regions where they fail to be linearly independent. We identify these regions although we have not been able in a mathematically rigorous fashion to describe the structure of the associated invariant submanifolds. Nevertheless, the phase space trajectories contained in these submanifolds

Hierarchy of integrable geodesic flows

Publicacions Matemàtiques, 2000

A family of integrable geodesic flows is obtained. Any such a family corresponds to a pair of geodesically equivalent metrics.

Rational integrals of 2-dimensional geodesic flows: New examples

Journal of Geometry and Physics, 2021

This paper is devoted to searching for Riemannian metrics on 2surfaces whose geodesic flows admit a rational in momenta first integral with a linear numerator and denominator. The explicit examples of metrics and such integrals are constructed. Few superintegrable systems are found having both a polynomial and a rational integrals which are functionally independent of the Hamiltonian.

An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach

Applied and Computational Harmonic Analysis, 2007

Here shape space is either the manifold of simple closed smooth unparameterized curves in R 2 or is the orbifold of immersions from S 1 to R 2 modulo the group of diffeomorphisms of S 1 . We investige several Riemannian metrics on shape space: L 2 -metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two length-weighted metrics. Sobolev metrics of order n on curves are described. Here the horizontal projection of a tangent field is given by a pseudo-differential operator. Finally the metric induced from the Sobolev metric on the group of diffeomorphisms on R 2 is treated. Although the quotient metrics are all given by pseudo-differential operators, their inverses are given by convolution with smooth kernels. We are able to prove local existence and uniqueness of solution to the geodesic equation for both kinds of Sobolev metrics.

Geodesic flows in manifolds of nonpositive curvature

Smooth Ergodic Theory and Its Applications, 2001

Introduction-a quick historical survey of geodesic flows on negatively curved spaces. II. Preliminaries on Riemannian manifolds A. Riemannian metric and Riemannian volume element B. Levi Civita connection and covariant differentiation along curves C. Parallel translation of vectors along curves D. Curvature E. Geodesics and geodesic flow F. Riemannian exponential map and Jacobi vector fields G. Isometries and local isometries H. Geometry of the tangent bundle with the Sasaki metric III. Manifolds of nonpositive sectional curvature A. Definition of nonpositive curvature by triangle comparisons B. Growth of Jacobi vector fields C. The Riemannian exponential map is a covering map. Theorem of Cartan-Hadamard. D. Examples : Riemannian symmetric spaces E. Convexity properties and the Cartan Fixed Point Theorem F. Fundamental group of a nonpositively curved manifold. G. Rank of a nonpositively curved manifold IV. Sphere at infinity of a simply connected manifold of nonpositive sectional curvature A. Asymptotic geodesics and cone topology for M (∞) B. Busemann functions and horospheres _______________________________________ Supported in part by NSF Grant DMS-9625452 2 C. Extension of isometries to homeomorphisms of the sphere at infinity. D. Relating the action of the geodesic flow of M on T 1 M to the action of π 1 (M) on M (∞) V. Measures on the sphere at infinity A. Harmonic measures {ν p : p ∈ M } B. Patterson-Sullivan measures {µ p : p ∈ M } C. Lebesgue measures {λ p : p ∈ M } D. Barycenter map for probability measures. VI. Anosov foliations in the unit tangent bundle T 1 M A. Stable and unstable Jacobi vector fields. B. The stable and unstable foliations E s and E u in T(T 1 M) C. The strong stable and strong unstable foliations E ss and E uu in T(T 1 M). D. Conditions for the foliations E ss and E uu to be Anosov. VII. Some outstanding problems of geometry and dynamics A. The Katok entropy conjecture B. Smoothness of Anosov foliations and Riemannian symmetric spaces C. The geodesic conjugacy problem D. Harmonic and asymptotically harmonic spaces E. Early partial solutions. VIII. The work of Besson-Courtois-Gallot A. Statement of the main result. B. Corollaries of the main result. C. Sketch of the proof of the main result. IX. References I.