Integrable Geodesic Flows Research Articles

Killing tensor fields on the 2-torus

A symmetric tensor field on a Riemannian manifold is called a Killing field if the symmetric part of its covariant derivative equals zero. There is a one-to-one correspondence between Killing tensor fields and first integrals of the geodesic flow which depend polynomially on the velocity. Therefore Killing tensor fields relate closely to the problem of integrability of geodesic flows. In particular, the following question is still open: does there exist a Riemannian metric on the 2-torus which admits an irreducible Killing tensor field of rank ≥ 3? We obtain two necessary conditions on a Riemannian metric on the 2-torus for the existence of Killing tensor fields. The first condition is valid for Killing tensor fields of arbitrary rank and relates to closed geodesics. The second condition is obtained for rank 3 Killing tensor fields and pertains to isolines of the Gaussian curvature.

On an integrable magnetic geodesic flow on the two-torus

We completely integrate the magnetic geodesic flow on a flat two-torus with the magnetic field $F = \cos (x) dx \wedge dy$ and describe all contractible periodic magnetic geodesics. It is shown that there are no such geodesics for energy $E \geq 1/2$, for $E< 1/2$ simple periodic magnetic geodesics form two $S^1$-families for which the (fixed energy) action functional is positive and therefore there are no periodic magnetic geodesics for which the action functional is negative.

Integrable geodesic flows on 2-torus: Formal solutions and variational principle

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 Bialy and Mironov (2011) that this is a semi-Hamiltonian system and we show here that the metric associated with the system is a metric of Egorov type. We use this fact in order to prove that in the case of integrals of degree three and four the system is in fact equivalent to a single remarkable equation of order 3 and 4 respectively. Remarkably the equation for the case of degree four has variational meaning: it is Euler–Lagrange equation of a variational principle. Next we prove that this equation for n = 4 has formal double periodic solutions as a series in a small parameter.

Completely integrable 2D Lagrangian systems and related integrable geodesic flows on various manifolds

In this study we have formulated a theorem that generates deformations of the natural integrable conservative systems in the plane into integrable systems on Riemannian and other manifolds by introducing additional parameters into their structures. The relation of explicit solutions of the new and the original dynamics to the corresponding Jacobi (Maupertuis) geodesic flow is clarified. For illustration, we apply the result to three concrete examples of the many available integrable systems in the literature. Complementary integrals in those systems are polynomial in velocity with degrees 3, 4 and 6, respectively. As a special case of the first deformed system, a new several-parameter family of integrable mechanical systems (and geodesic flows) on S2 is constructed.

On a class of integrable systems with a quartic first integral

We generalize, to some extent, the results on integrable geodesic flows on two dimensional manifolds with a quartic first integral in the framework laid down by Selivanova and Hadeler. The local structure is first determined by a direct integration of the differential system which expresses the conservation of the quartic observable and is seen to involve a finite number of parameters. The global structure is studied in some details and leads to a class of models living on the manifolds S^2, H^2 or R^2. As special cases we recover Kovalevskaya's integrable system and a generalization of it due to Goryachev.

Weak integrability of Hamiltonians on the two torus and rigidity

We show that a C∞ k-basic Finsler metric on the two torus T2 whose geodesic flow preserves a codimension one C1,L foliation is in fact flat. Although integrable high energy levels of Hamiltonians on the torus are not flat in general, the C1,L integrability of k-basic Finsler geodesic flows on T2 implies flatness and in particular, C∞ integrability. We also show that a codimension one C1 foliation invariant by the geodesic flow of any Finsler metric on T2 must coincide with the Busemann foliation. A consequence of the above results is that the Hopf conjecture would be false for k-basic Finsler metrics on the two torus if and only if there exists a C0 integrable k-basic Finsler geodesic flow that is not C1,L integrable.

TWO REMARKS ON PQε-PROJECTIVITY OF RIEMANNIAN METRICS

AbstractWe show that PQε-projectivity of two Riemannian metrics introduced in [15] (P. J. Topalov, Geodesic compatibility and integrability of geodesic flows, J. Math. Phys.44(2) (2003), 913–929.) implies affine equivalence of the metrics unless ε ∈ {0,−1,−3,−5,−7,. . .}. Moreover, we show that for ε=0, PQε-projectivity implies projective equivalence.

Further results on non-diagonal: Bianchi type III vacuum metrics

We present the derivation, for these vacuum metrics, of the Painlev\'e VI equation first obtained by Christodoulakis and Terzis, from the field equations for both minkowskian and euclidean signatures. This allows a complete discussion and the precise connection with some old results due to Kinnersley. The hyperk\"ahler metrics are shown to belong to the Multi-Centre class and for the cases exhibiting an integrable geodesic flow the relevant Killing tensors are given. We conclude by the proof that for the Bianchi B family, excluding type III, there are no hyperk\"ahler metrics.

On the second term in the Weyl formula for the spectrum of the Laplace operator on the two-dimensional torus and the number of integer points in spectral domains

We construct Liouville metrics on the two-dimensional torus for which the asymptotic behaviour of the second term in the Weyl formula is evaluated explicitly. We prove the instability of the second term in this formula with respect to small deformations (in the metric) of a Liouville metric, and establish the absence of power reduction in the Hörmander estimate on the class of closed manifolds with smooth metric in the case of integrable geodesic flow and the zero measure of the set of closed geodesics in the subspace of unit spheres of the cotangent bundle.

Toric Integrable Geodesic Flows in Odd Dimensions

Let Q be a compact, connected n-dimensional Riemannian manifold, and assume that the geodesic flow is toric integrable. If n 6= 3 is odd, or if π1(Q) is infinite, we show that the cosphere bundle of Q is equivariantly contactomorphic to the cosphere bundle of the torus Tn. As a consequence, Q is homeomorphic to Tn.

Geodesic flows and Neumann systems on Stiefel varieties: geometry and integrability

We study integrable geodesic flows on Stiefel varieties $V_{n,r}=SO(n)/SO(n-r)$ given by the Euclidean, normal (standard), Manakov-type, and Einstein metrics. We also consider natural generalizations of the Neumann systems on $V_{n,r}$ with the above metrics and proves their integrability in the non-commutative sense by presenting compatible Poisson brackets on $(T^*V_{n,r})/SO(r)$. Various reductions of the latter systems are described, in particular, the generalized Neumann system on an oriented Grassmannian $G_{n,r}$ and on a sphere $S^{n-1}$ in presence of Yang-Mills fields or a magnetic monopole field. Apart from the known Lax pair for generalized Neumann systems, an alternative (dual) Lax pair is presented, which enables one to formulate a generalization of the Chasles theorem relating the trajectories of the systems and common linear spaces tangent to confocal quadrics. Additionally, several extensions are considered: the generalized Neumann system on the complex Stiefel variety $W_{n,r}=U(n)/U(n-r)$, the matrix analogs of the double and coupled Neumann systems.

Rich quasi-linear system for integrable geodesic flows on 2-torus

Consider a Riemannian metric on two-torus. We prove that thequestion of existence of polynomial first integrals leads naturallyto a remarkable system of quasi-linear equations which turns out tobe a Rich system of conservation laws. This reduces the question ofintegrability to the question of existence of smooth (quasi-)periodic solutions for this Rich quasi-linear system.

On Liouville integrability of $h$-projectively equivalent Kähler metrics

Under a nondegeneracy condition we classify the compact connected Kahler manifolds admitting pairs of h-projectively equivalent metrics. Any such manifold is biholomorphically equivalent to ℂP n and has integrable geodesic flow.

Integrable Geodesic Flows on Surfaces

We propose a new condition $${{\aleph}}$$ which enables us to get new results on integrable geodesic flows on closed surfaces. This paper has two parts. In the first, we strengthen Kozlov’s theorem on non-integrability on surfaces of higher genus. In the second, we study integrable geodesic flows on a 2-torus. Our main result for a 2-torus describes the phase portraits of integrable flows. We prove that they are essentially standard outside what we call separatrix chains. The complement to the union of the separatrix chains is C 0-foliated by invariant sections of the bundle.

Geodesic Vlasov equations and their integrable moment closures

Various integrable geodesic flows on Lie groups are shown to arise by taking moments of a geodesic Vlasov equation on the group of canonical transformations. This was already known for both the one- and two-component Camassa-Holm systems [18, 19]. The present paper extends our earlier work to recover another integrable system of ODE's that was recently introduced by Bloch and Iserles [5]. Solutions of the Bloch-Iserles system are found to arise from the Klimontovich solution of the geodesic Vlasov equation. These solutions are shown to form one of the legs of a dual pair of momentum maps. The Lie-Poisson structures for the dynamics of truncated moment hierarchies are also presented in this context.

Integrable magnetic geodesic flows on lie groups

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 condition of integrability in quadratures is formulated. Canonic forms for 2-cocycles of all 4-dimensional Lie algebras are given and integrable cases among them are separated.

Topology of real algebraic curves and integrability of geodesic flows on algebraic surfaces

Topology of real algebraic curves and integrability of geodesic flows on algebraic surfaces

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.

Integrable geodesic flows and multi-centre versus Bianchi A metrics

It is shown that most, but not all, of the four-dimensional metrics in the multi-centre family with integrable geodesic flows may be recognized as belonging to spatially homogeneous Bianchi type A metrics. We show that any diagonal bi-axial Bianchi II metric has an integrable geodesic flow, and that the simplest metric in this family displays a finite-dimensional W-algebra for its observables. Our analysis also sheds light on a non-diagonal Bianchi VII0 metric which seems to be new. We conclude by showing that the elliptic coordinates advocated in the literature do not separate the Hamilton–Jacobi equation for the tri-axial Bianchi IX metric.

An optical Hamiltonian and obstructions to integrability

If the geodesic flow of a compact Finslerian 3-manifold is completely integrable, and its singular set is a tame polyhedron, then the manifold's π1 is almost solvable (Butler 2005 Topology 44 769–89). Is this true for non-commutatively integrable geodesic flows? This paper constructs non-commutatively integrable optical Hamiltonians on the unit-sphere bundle of homogeneous spaces of PSL2R that have a real-analytic singular set. These flows are not tangential to a Lagrangian foliation with a tame singular set.