Related papers: Quadratically integrable geodesic flows on the tor…

We study the geodesic flow on the global holomorphic sections of the bundle $\pi:{TS}^2\to {S}^2$ induced by the neutral K\"ahler metric on the space of oriented lines of ${\Bbb{R}}^3$, which we identify with ${TS}^2$. This flow is shown to…

Let (T^2, g) be a two-dimensional Riemannian torus. In this paper we prove that the topological entropy of the geodesic flow restricted to the set of initial conditions of minimal geodesics vanishes, independent of the choice of the…

We prove the integrability of geodesic flows on the Riemannian g.o. spaces of compact Lie groups, as well as on a related class of Riemannian homogeneous spaces having an additional principal bundle structure.

Right-invariant geodesic flows on manifolds of Lie groups associated with 2-cocycles of corresponding Lie algebras are discussed. Algebra of integrals of motion for magnetic geodesic flows is considered and necessary and sufficient…

We discuss canonical transformations relating well-known geodesic flows on the cotangent bundle of the sphere with a set of geodesic flows with quartic invariants. By adding various potentials to the corresponding geodesic Hamiltonians, we…

We use a theorem of P. Berger and D. Turaev to construct an example of a Finsler geodesic flow on the 2-torus with a transverse section, such that its Poincar\'e return map has positive metric entropy. The Finsler metric generating the flow…

In this article we investigate the dynamical properties of the geodesic flow for a proper metric space endowed with a proper action by isometries of a group with a contracting element. We show that the existence of a contracting isometry is…

We describe a class of completely integrable $G$-invariant magnetic geodesic flows on (co)adjoint orbits of a compact connected Lie group $G$ with magnetic field given by the Kirillov-Konstant 2-form.

In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus $T^2$ for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the…

Let $Q$ be a closed manifold admitting a locally-free action of a compact Lie group $G$. In this paper we study the properties of geodesic flows on $Q$ given by Riemannian metrics which are invariant by such an action. In particular, we…

This paper addresses an open problem recently posed by V. Kozlov: a rigorous proof of the non-integrability of the geodesic flow on the cubic surface $x y z=1$. We prove this is the case using the Morales-Ramis theorem and Kovacic…

We prove that a surface carries a hexagonal 3-web of geodesics if and only if the geodesic flow on the surface admits a cubic first integral and show that the system of partial differential equations, governing metrics on such surfaces, is…

In this paper we study quasi-linear system of partial differential equations which describes the existence of the polynomial in momenta first integral of the integrable geodesic flow on 2-torus. We proved in [3] that this is a…

We explicitely construct an example of an analytic metric on $T^2$ which is non-separable but it is locally integrable on an energy surface. The construction is based on a KAM-like approach and a careful control on what happens on the…

We consider a transversally conformal foliation $\mathcal{F}$ of a closed manifold $M$ endowed with a smooth Riemannian metric whose restriction to each leaf is negatively curved. We prove that it satisfies the following dichotomy. Either…

We prove that the geodesic flow for the Weil-Petersson metric on the moduli space of Riemann surfaces is ergodic (in fact Bernoulli) and has finite, positive metric entropy.

In this paper, we study visibility representations of graphs that are embedded on a torus or a Klein bottle. Mohar and Rosenstiehl showed that any toroidal graph has a visibility representation on a flat torus bounded by a parallelogram,…

In this work we study the geodesic flow on nilmanifolds associated to graphs. We are interested in the construction of first integrals to show complete integrability on some compact quotients. Also examples of integrable geodesic flows and…

We consider magnetic geodesic flows of the normal metrics on a class of homogeneous spaces, in particular (co)adjoint orbits of compact Lie groups. We give the proof of the non-commutative integrability of flows and show, in addition, for…

We show that the motion on the n-dimensional ellipsoid is complete integrable by exhibiting n integrals in involution. The system is separable at classical and quantum level, the separation of classical variables being realized by the…