Geometrically interprets integrability of geodesic flow using web theory.
problem Integrability of geodesic flow by quadratic integrals.
method Geometric interpretation through web theory and integrable billiards construction.
result Constructs integrable billiards on surfaces with quadratic geodesic integrals.
Geodesic flows on surfaces have specific fractional-linear integrals related to constant cross-ratios.
problem Characterizing geodesic flows on surfaces with fractional-linear integrals.
method Proving the dimension of fractional-linear integrals and giving a geometric criterion.
result The dimension of fractional-linear integrals is either 3 or 5, corresponding to constant curvature.
Geodesic flows with specific integrals are linked to special 4-webs.
problem Characterizing geodesic flows with commuting quadratic integrals.
method Characterization through geodesic 4-webs and geometric hypothesis.
result Metrics of Stäckel type for geodesic flows with specific integrals.
Study integrability of conformal geodesics on gravitational instantons.
problem Integrability of conformal geodesic flow on gravitational instantons.
method Analyzing conformal geodesic flow equations on SO(3)-invariant gravitational instantons, separating Hamilton-Jacobi equations, finding commuting first integrals, and using conformal Killing-Yano tensors. result First example of an integrable conformal geodesic flow on a non-symmetric four-manifold.
Geodesic flows with diagonalisable integrals are orthogonal.
problem Understanding geodesic flows with specific integrals.
method Analyzing quadratic integrals for geodesic flows.
result Diagonalisable integrals imply orthogonal separation of variables.
Geodesically equivalent Finsler metrics share invariant volume forms and first integrals.
problem Understanding shared properties of geodesically equivalent Finsler metrics.
method Computing first integrals as coefficients of a characteristic polynomial.
result Geodesically invariant functions are first integrals of geodesically equivalent Finsler metrics.
Study geodesic flows on cones over Riemannian manifolds, showing superintegrability.
problem Behavior of geodesics on cones over arbitrary Riemannian manifolds.
method Show existence of first integrals uniquely determining geodesics.
result Geodesic flow on cones is superintegrable and Liouville--Arnold integrable for non-radial trajectories.
The paper finds new metrics for geodesic flows with rational integrals.
problem Finding Riemannian metrics with rational integrals for geodesic flows.
method Explicit construction of metrics and integrals.
result New examples of metrics with rational integrals are provided.
Study integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
problem Integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
method Semi-Hamiltonian systems of PDEs and generalized hodograph method.
result Construction of many local explicit and implicit integrable examples with polynomial first integrals of degrees 3, 4, 5.
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 of non-integrable ones are shown.
We suggest a construction that, given a trajectorial diffeomorphism between two Hamiltonian systems, produces integrals of them. As the main example we treat geodesic equivalence of metrics. We show that the existence of a non-trivially geodesically equivalent metric leads to Liouville integrability, and present explic…
Relates geodesic integrals to Killing tensors, exploring their dimensions.
problem Understanding geodesic integrals and Killing tensors in Riemannian geometry.
method Relating rational integrals of geodesic flow to relative Killing tensors, analyzing their span and dimensions.
result Upper bounds on dimensions of spaces spanned by these integrals and tensors.
Proves magnetic geodesic flow on sphere is integrable with constant 2-form.
problem Proving integrability of magnetic geodesic flow on sphere.
method Analyzes magnetic geodesic flow on sphere with constant 2-form.
result Proves Liouville integrability of magnetic geodesic flow on sphere.
The study provides a criterion for fractional-linear integrals of geodesics on surfaces.
problem Existence and classification of fractional-linear integrals for geodesic flows on Riemannian surfaces.
method Criterion and analysis of moduli space of local integrals.
result The moduli space of such local integrals is either the 2D projective plane or finite points.
The geodesic flow of a Riemannian metric on a compact manifold Q is said to be toric integrable if it is completely integrable and the first integrals of motion generate a homogeneous torus action on the punctured cotangent bundle T∗Q∖Q. If the geodesic flow is toric integrable, the cosphere bundle admit…
Integrable geodesics found on special orthogonal group.
problem Analyzing normal geodesics on the special orthogonal group.
method Adapted Lax pair and bi-Hamiltonian structure.
result Almost all normal geodesics are completely integrable.
The study describes metrics geodesically compatible with Nijenhuis operators and their applications to integrable systems.
problem Geodesically compatible metrics and their applications to integrable systems.
method Describes metrics geodesically compatible with a gl-regular Nijenhuis operator and shows how these metrics relate to integrable PDE systems.
result Every metric geodesically compatible with a Nijenhuis operator gives a finite-dimensional reduction of an integrable PDE system.
By studying completely integrable torus actions on contact manifolds we prove a conjecture of Toth and Zelditch that toric integrable geodesic flows on tori must have flat metrics.
For any toric automorphism with only real eigenvalues a Riemannian metric with an integrable geodesic flow on the suspension of this automorphism is constructed. A qualitative analysis of such a flow on a three-solvmanifold constructed by the authors in math.DG/9905078 is done. This flow is an example of the geodesic f…
A vector field on a Riemannian manifold is called geodesic if its integral curves are reparametrized geodesics. We classify compact Kähler manifolds admitting nontrivial real-holomorphic geodesic gradient vector fields that satisfy an additional integrability condition. They are all biholomorphic to bundles of complex …
The problem of the existence of an additional (independent on the energy) first integral, of a geodesic (or magnetic geodesic) flow, which is polynomial in momenta is studied. The relation of this problem to the existence of nontrivial solutions of stationary dispersionless limits of two-dimensional soliton equations i…
In the present paper we prove, that if the geodesic flow of a metric G on the torus T is quadratically integrable, then the torus T isometrically covers a torus with a Liouville metric on it, and describe the set of quadratically integrable geodesic flows on the Klein bottle.
Partial regularity theorem for geodesic networks in R^n.
problem Proving regularity for geodesic networks in R^n.
method Partial ε-regularity theorem for sequences of integer multiplicity stationary geodesic networks converging to a multiplicity k line. result Proves partial regularity for geodesic networks.
Geodesic algorithms extended to arbitrary ellipsoids.
problem Computing geodesics on ellipsoids of varying eccentricity.
method Implementation of geodesic algorithms using elliptic integrals and discrete sine transform.
result Achieved high accuracy (close to machine precision) for geodesic computations.
Study on integrability of geodesic flows on Heisenberg group.
problem Integrability of geodesic flows on Heisenberg group.
method Investigation of two classes of normal geodesic flows associated with left-invariant sub-Riemannian metric.
result Left-left configuration is completely integrable in non-commutative sense, while left-right configuration exhibits non-commutative integrability in dimensions > 5.
We study n-dimensional Kähler manifolds whose geodesic flows possess n first integrals in involution that are fibrewise hermitian forms and simultaneously normalizable. Under some mild assumption, one can associate with such a manifold an n-dimensional commutative Lie algebra of infinitesimal automorphisms. This,…
Complete integrability proved for SR geodesic flow on S^7.
problem Complete integrability of subriemannian geodesic flow on S^7.
method Adapting a method by A. Thimm, constructing functionally independent first integrals.
result Complete integrability in the sense of Liouville proved for SR geodesic flow.
The paper studies magnetic geodesic flows on 2-surfaces with integrable structures.
problem Analyzing magnetic geodesic flows on 2-surfaces with additional integrals.
method Constructing exact solutions to semi-Hamiltonian systems of PDEs using generalized hodograph method and Legendre transformation.
result Exact solutions constructed for semi-Hamiltonian systems of PDEs.
Jacobi solved geodesics on triaxial ellipsoids.
problem Finding the shortest path on a triaxial ellipsoid.
method Numerical evaluation of integrals and solving coupled equations.
result Solution for geodesics on triaxial ellipsoids.
Formula for integrating random variables on hyperbolic surfaces.
problem Integrating random variables on the moduli space of hyperbolic surfaces.
method Integration formula for lengths of closed geodesics.
result Integral of geometric random variables can be expressed as an integral over R.
Study reformulates Finsler metrizability problems using geodesic invariance.
problem Finsler metrizability problems for sprays.
method Reformulate problems in terms of geodesic invariance of tensors (metric and angular).
result Gyroscopic sprays have geodesically invariant angular metric.
Study extends geodesic curvature formula to higher dimensions.
problem Extending curvature formula to higher-dimensional spheres.
method Using new integral-geometric formulas for Euclidean and geodesic total curvature.
result Explicit formula for geodesic total curvature on higher-dimensional spheres.
We show that an invariant surface allows to construct the Jacobi vector field along a geodesic and construct the formula for the normal component of the Jacobi field. If a geodesic is the transversal intersection of two invariant surfaces (such situation we have, for example, if the geodesic is hyperbolic), then we can…
We propose a new condition ℵ which enables 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 2-torus. Our main result for…
Any smooth geodesic flow is locally integrable with smooth integrals. We show that generically this fails if we require, in addition, that the integrals are polynomial (or, more generally, analytic) in momenta. Consequently we obtain that a generic real-analytic metric does not admit, even locally, a real-analytic inte…
The paper proves geodesibility for algebrizable 3D vector fields.
problem Establishing geodesibility for algebrizable 3D vector fields.
method Rectifications and Riemannian metrics are provided for the vector fields, and two first integrals are given.
result Geodesibility is proven for algebrizable 3D vector fields.
Proposes Geodesic Integrated Gradients (GIG) for more accurate feature attributions in deep networks.
problem Flawed attributions using straight paths from Integrated Gradients (IG).
method Introduces a model-induced Riemannian metric and computes attributions along geodesics.
result GIG produces more faithful attributions than IG on benchmarks.
Study finds homogeneous spaces with geodesic orbits but no integrable distributions.
problem Characterizing homogeneous spaces with specific geometric properties.
method Examined Lie groups with compact stabilizers and classified spaces based on geodesic orbits and integrable distributions.
result Identified homogeneous spaces with geodesic orbits but lacking integrable invariant distributions.
Geodesic flows on Kähler manifolds are quantum integrable when metrics are c-projectively equivalent.
problem Quantum integrability of geodesic flows on Kähler manifolds under c-projective equivalence.
method Construction of Poisson-commuting integrals of motion and their quantum counterparts.
result The geodesic flow's integrals of motion commute as quantum operators, leading to separation of variables in Schrödinger's equation.
An example of a real-analytic metric on a compact manifold whose geodesic flow is Liouville integrable by C∞ functions and has positive topological entropy is constructed.
If (M,g) is a compact Riemannian surface then the integrals of L2(M)-normalized eigenfunctions ej over geodesic segments of fixed length are uniformly bounded. Also, if (M,g) has negative curvature and γ(t) is a geodesic parameterized by arc length, the measures ej(γ(t))dt on R tend to zero in the …
Reconstructs piecewise constant functions from geodesic integrals.
problem Recovering piecewise constant functions from X-ray data.
method Injectivity proof using variations through geodesics, improved for simple manifolds.
result Explicit formulas for function values near the boundary and stability analysis.
Integrates magnetic geodesic and sub-Riemannian flows on Stefel variety, proving integrability and Lax presentations.
problem Integrability of magnetic geodesic and sub-Riemannian flows on Vn,2. method Proves integrability of magnetic geodesic and sub-Riemannian flows on Vn,2 with respect to magnetic field ηdα. result Integrable cases of a heavy rigid body with a gyrostat are derived.
Given a compact manifold with boundary with unknown Riemannian metric. The problem is to reconstruct the metric in a class of conformal metrics from knowledge of lengths of all closed geodesics (kinematic data). An integral inequality is stated which implies uniqueness and stability for this problem. If the conformal c…
The behavior of geodesic curves on even seemingly simple surfaces can be surprisingly complex. In this paper we use the Hamiltonian formulation of the geodesic equations to analyze their integrability properties. In particular, we examine the behavior of geodesics on surfaces defined by the spherical harmonics. Using t…
Conditions for metallic structures to induce integrable distributions and geodesic invariance.
problem Conditions for metallic structures to induce integrable and geodesically invariant distributions.
method Analyzing tensor fields and adapted connections associated with metallic structures.
result Chen-type inequality for metallic distributions and conditions for metallic maps to preserve these distributions.
Stable nets on convex hypersurfaces maintain their shape under small perturbations.
problem Maintaining the shape of nets on convex surfaces under slight changes.
method Constructing stable geodesic nets on convex hypersurfaces.
result Stable geodesic nets on convex hypersurfaces do not change shape under small perturbations.
It has been proved that on 2-dimensional orientable compact manifolds of genus g>1 there is no integrable geodesic flow with an integral polynomial in momenta. There is a conjecture that all integrable geodesic flows on T2 possess an integral quadratic in momenta. All geodesic flows on S2 and T2 possessing i…