A topological classification of integrable geodesic flows on the two-dimensional sphere with an additional integral quadratic in the momenta

Summary The analytic expression for a Riemannian metric on a 2-sphere, having integrable geodesic flow with an additional integral quadratic in momenta, is given in [Ko1]. We give the topological classification, up to topological equivalence of Liouville foliations, of all such metrics. The classification is computable, and the formula for calculating the complexity of the flow is straightforward. We prove Fomenko's conjecture that, from the point of view of complexity, the integrable geodesic flows with an additional integral linear or quadratic in momenta exhaust “almost all” integrable geodesic flows on the 2-dimensional sphere.     

Integrability of geodesic flows and isospectrality of Riemannian manifolds

We construct a pair of compact, eight-dimensional, two-step Riemannian nilmanifolds M and M′ which are isospectral for the Laplace operator on functions and such that M has completely integrable geodesic flow in the sense of Liouville, while M′ has not. Moreover, for both manifolds we analyze the structure of the submanifolds of the unit tangent bundle given by two maximal continuous families of closed geodesics with generic velocity fields. The structure of these submanifolds turns out to reflect the above (non)integrability properties. On the other hand, their dimension is larger than that of the Lagrangian tori in M, indicating a degeneracy which might explain the fact that the wave invariants do not distinguish an integrable from a nonintegrable system here. Finally, we show that for M, the invariant eight-dimensional tori which are foliated by closed geodesics are dense in the unit tangent bundle, and that both M and M′ satisfy the so-called Clean Intersection Hypothesis. The author was partially supported by DFG Sonderforschungsbereich 647.     

Topological entropy for geodesic flows on fibre bundles over rationally hyperbolic manifolds

Let be the total space of a fibre bundle with base a simply connected manifold whose loop space homology grows exponentially for a given coefficient field. Then we show that for any Riemannian metric on , the topological entropy of the geodesic flow of is positive. It follows then, that there exist closed manifolds with arbitrary fundamental group, for which the geodesic flow of any Riemannian metric on has positive topological entropy.

     

Manifolds of infinite topological type with integrable geodesic flows

Let (M,k) be a complete surface of constant negative curvature (resp. an -geometric 3-manifold). This paper constructs a complete riemannian 8-manifold (resp. 9-manifold) (,h) such that is homotopy equivalent to M, the geodesic flow of h is completely integrable and there is a riemannian embedding (M,k)(,h). This embeds the geodesic flow of (M,k) as a subsystem of an integrable geodesic flow. Amongst the manifolds is an 8-dimensional manifold whose fundamental group is the free group on countably many generators.Thanks to Keith Burns and Leo Jonker for comments. Research partially supported by the Natural Sciences and Engineering Research Council of Canada.Mathematics Subject Classification (2000): 58F17, 53D25, 37D40     

Liouville classification of integrable geodesic flows on a torus of revolution in a potential field

A Liouville classification of integrable Hamiltonian systems being geodesic flows on a twodimensional torus of revolution in an invariant potential field is obtained in the case of linear integral. This classification is obtained using the Fomenko–Zieschang invariant (so called marked molecules) of the systems under consideration. All types of bifurcation curves are described. A classification of singularities of the system solutions is also obtained.     

Topological structure of complete Riemannian manifolds with cyclic holonomy groups

Let M be a complete m-dimensional Riemannian manifold with cyclic holonomy group, let X be a closed flat manifold homotopy equivalent to M, and let L→X be a nontrivial line bundle over X whose total space is a flat manifold with cyclic holonomy group. We prove that either M is diffeomorphic to X×Rm-dimX or M is diffeomorphic to L×Rm-dimX−1.     

Orbital classification of geodesic flows on two-dimensional ellipsoids. The Jacobi problem is orbitally equivalent to the integrable Euler case in rigid body dynamics

Moscow State University, Department of Mathematics and Mechanics. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 29, No. 3, pp. 1–15, July–September, 1995.     

Entropy-expansiveness of geodesic flows on closed manifolds without conjugate points

In this article, we consider the entropy-expansiveness of geodesic flows on closed Riemannian manifolds without conjugate points. We prove that, if the manifold has no focal points, or if the manifold is bounded asymptote, then the geodesic flow is entropy-expansive. Moreover, for the compact oriented surfaces without conjugate points, we prove that the geodesic flows are entropy-expansive. We also give an estimation of distance between two positively asymptotic geodesics of an uniform visibility manifold.     

The structure of integrable Lax flows on nonlocal manifolds: Dynamic systems with sources

We consider a dynamic system on the extended phase space to the initial Lie algebra and study its generalized Hamiltonian and integrability in the cases when the initial Lie algebra coincides with the Grassmann algebra of pseudodifferential operators on the real line and on the centrally extended affine Lie algebra.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 106–115.     

On the Pesin set of expansive geodesic flows in manifolds with no conjugate points

In this paper, we show that the Pesin set of an expansive geodesic flow in compact manifold with no conjugate points and bounded asymptote coincides a.e with an open and dense set of the unit tangent bundle. We also show that the set of hyperbolic periodic orbits is dense in the unit tangent bundle.     

The limit spaces of two-dimensional manifolds with uniformly bounded integral curvature

We study the class of closed -dimensional Riemannian manifolds with uniformly bounded diameter and total absolute curvature. Our first theorem states that this class of manifolds is precompact with respect to the Gromov-Hausdorff distance. Our goal in this paper is to completely characterize the topological structure of all the limit spaces of the class of manifolds, which are, in general, not topological manifolds and even may not be locally -connected. We also study the limit of -manifolds with -curvature bound for .

     

Exact smooth classification of hamiltonian vector fields on two-dimensional manifolds

A complete exact classification of Hamiltonian systems with Morse Hamiltonians on two-dimensional manifolds is given, i.e., the systems are classified up to diffeomorphisms mapping vector fields into vector fields. The classification imposes no restrictions on Morse functions. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 179–200, February, 1997. Translated by O. V. Sipacheva     

Exact topological classification of Hamiltonian flows on smooth two-dimensional surfaces

The present paper contains an exact topological classification of all nondegenerate Hamiltonian systems on smooth closed two-dimensional surfaces. Bibliography: 8 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 235, 1900, pp. 22–53.     

Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural?

It is shown that the geometrically correct investigation of regularity of nonlinear differential flows on manifolds and related parabolic equations requires the introduction of a new type of variations with respect to the initial data. These variations are defined via a certain generalization of a covariant Riemannian derivative to the case of diffeomorphisms. The appearance of curvature in the structure of high-order variational equations is discussed and a family of a priori nonlinear estimates of regularity of any order is obtained. By using the relationship between the differential equations on manifolds and semigroups, we study C ^∞-regular properties of solutions of the parabolic Cauchy problems with coefficients increasing at infinity. The obtained conditions of regularity generalize the classical coercivity and dissipation conditions to the case of a manifold and correlate (in a unified way) the behavior of diffusion and drift coefficients with the geometric properties of the manifold without traditional separation of curvature. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1011–1034, August, 2006.