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.
Two metrics on a manifold are geodesically equivalent if sets of their unparameterized geodesics coincide. In this paper we show that if two left G-invariant metrics of arbitrary signature on homogenous space G/H are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connecti…
We study the geodesics on an invariant surface of a three dimensional Riemannian manifold. The main results are: the characterization of geodesic orbits; a Clairaut's relation and its geometric interpretation in some remarkable three dimensional spaces; the local description of the geodesics; the explicit description o…
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.
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.
We provide an easy approach to the geodesic distance on the general linear group GL(n) for left-invariant Riemannian metrics which are also right-O(n)-invariant. The parametrization of geodesic curves and the global existence of length minimizing geodesics are deduced using simple methods based on the calculus of varia…
In this paper we show that for an invariant (α,β)−metric F on a homogeneous Finsler manifold HG, induced by an invariant Riemannian metric a~ and an invariant vector field X~, the vector X=X~(H) is a geodesic vector of F if and only if it is a geodesic vector of a~. …
Geodesic orbit metrics on real flag manifolds identified.
problem Classifying real flag manifolds with geodesic orbit metrics.
method Investigated invariant metrics on real flag manifolds, focusing on those where geodesics are orbits of one-parameter subgroups.
result Non-trivial geodesic orbit metrics exist on real flag manifolds, unlike in the complex case.
We describe the geometry of geodesics on a Lorentz ellipsoid: give explicit formulas for the first integrals (pseudo-confocal coordinates), curvature, geodesically equivalent Riemannian metric, the invariant area-forms on the time- and space-like geodesics and invariant 1-form on the space of null geodesics. We prove a…
Spray-invariant sets maintain geodesics on infinite-dimensional manifolds.
problem Geodesic preservation in infinite-dimensional manifolds.
method Definition of spray-invariant sets and analysis of their properties.
result Different geometric properties of spray-invariant sets based on their regularity.
In this paper, we study the set of homogeneous geodesics of a leftinvariant Finsler metric on Lie groups. We first give a simple criterion that characterizes geodesic vectors. As an application, we study some geometric properties of bi-invariant Finsler metrics on Lie groups. In particular a necessary and sufficient co…
Geodesics in Sol geometry described with invariant k and spiral properties.
problem Understanding the geodesic flow in the Sol geometry.
method Self-contained geometric description and analysis of geodesics.
result Characterization of geodesic segments, cut locus, and asymptotic distance growth.
Geodesic graphs for special Finsler metrics on spheres are studied.
problem Characterizing geodesic orbit Finsler metrics on spheres.
method Explicit constructions and group extensions.
result Not all projective spaces admit invariant Finsler metrics.
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…
Study curves on a Whitney umbrella using geometric invariants.
problem Analyzing curves passing through a specific geometric shape.
method Using Darboux frame and Frenet-Serre formulas, define invariants and investigate their properties.
result Identified degrees of divergence and top-terms of invariants.
New variational principles found for conformal geodesics.
problem Challenges in Lagrangian formulation for conformal geodesics.
method Enlarging the class of variations leads to a variational formulation with a third-order conformally invariant Lagrangian.
result Some integral curves of the fourth-order ODE system are spirals.
Study of Gödel Universe as Lie group with specific metric.
problem Characterize geodesics in the Gödel Universe.
method Geometric theory of optimal control applied to Lie groups with left-invariant Lorentz metrics.
result No closed timelike or isotropic geodesics in the Gödel Universe.
Normal geodesic flows flows of Carnot-Caratheodory are discussed from the point of view of the theory of Hamiltonian systems. The geodesic flows corresponding to left-invariant metrics and left- and -right-invariant rank 2 distributions on the three-dimensional Heisenberg group are analysed as integrable systems. The f…
Classifies left invariant Kundt structures on 3D Lie groups.
problem Understanding Kundt spacetimes and their properties.
method Analyzes local structure and properties of left invariant Kundt structures.
result Classifies all left invariant Kundt structures on 3D simply connected unimodular Lie groups.
Invariance principle proved for lifted geodesic walks on Riemannian submersions.
problem Proving convergence to horizontal Brownian motion for lifted geodesic walks.
method Appropriate conditions on geodesic random walks' speed; proving invariance principle.
result Convergence to horizontal Brownian motion for lifted geodesic walks.
We prove that on closed Riemannian manifolds with infinite abelian, but not cyclic, fundamental group, any isometry that is homotopic to the identity possesses infinitely many invariant geodesics. We conjecture that the result remains true if the fundamental group is infinite cyclic. We also formulate a generalization …
Study Einstein Lie groups and geodesic orbit manifolds, finding some are not geodesic orbit.
problem Characterizing Einstein Lie groups and geodesic orbit manifolds.
method Characterizing GimesK-invariant geodesic orbit metrics on Lie groups G for regular subgroups K. result Extensive classes of compact simple Einstein Lie groups are not geodesic orbit manifolds.
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.
In this paper, we investigate left-invariant geodesic orbit metrics on connected simple Lie groups, where the metrics are formed by the structures of generalized flag manifolds. We prove that all these left-invariant geodesic orbit metrics on simple Lie groups are naturally reductive.
Proves geodesic connections on 2-torus without invariant tori.
problem Existence of geodesic connections on 2-torus without invariant tori.
method Uses J. Mather's result on connecting orbits for monotone twist maps.
result Proves existence of connecting geodesics on unit tangent bundle of 2-torus.
We studied rules of transformations of Christoffel symbols under third type almost geodesic mappings in this paper. From this research, we obtained some new invariants of these mappings. These invariants are analogies of Thomas projective parameter and Weyl projective tensor.
Study left invariant spray structures on Lie groups, calculating curvature and geodesics.
problem Understanding curvature and geodesics in left invariant spray structures on Lie groups.
method Use invariant frames and canonical bi-invariant Berwald spray structure to analyze left invariant spray structures.
result Established correspondence between geodesics and inverse integral curves of spray vector fields.
Study magnetic geodesics on Heisenberg groups and manifolds.
problem Dynamics of magnetic flows on Heisenberg groups.
method Explicit description of magnetic geodesics, determination of lengths.
result Density of periodic magnetic geodesics and marked magnetic length spectrum rigidity.
Study magnetic geodesics on half-Lie groups, proving Hopf-Rinow theorem for energies above critical value.
problem Investigate magnetic geodesics on half-Lie groups using Riemannian and two-form structures.
method Define Mañé's critical value, prove Finsler geodesic flow equivalence, and apply Hopf-Rinow theorem.
result Hopf-Rinow theorem holds for energies above Mañé's critical value on magnetic geodesics.
A topological invariant of the geodesic laminations on a modular surface is constructed. The invariant has a continuous part (the tail of a continued fraction) and a combinatorial part (the singularity data). It is shown, that the invariant is complete, i.e. the geodesic lamination can be recovered from the invariant. …
Study Froyshov invariants and closed geodesics in hyperbolic 3-manifolds.
problem Understanding Froyshov invariants in hyperbolic 3-manifolds.
method Analytic number theory and hyperbolic geometry.
result Effective upper bounds for Froyshov invariants and algorithm to compute them.
Minimal number of geodesics in Finsler manifolds with indefinite Killing form is at least four.
problem Determining the minimal number of homogeneous geodesics in Finsler manifolds with indefinite Killing form.
method Analyzing examples of Lie groups with invariant Finsler metrics and presenting new examples.
result Homogeneous Finsler manifolds with indefinite Killing form admit at least four homogeneous geodesics.
Study Clairaut anti-invariant submersions on nearly Kaehler manifolds.
problem Characterize geometric properties of Clairaut anti-invariant submersions.
method Investigate conditions for total geodesic maps and totally umbilical fibers.
result Conditions for Clairaut anti-invariant submersions to be totally geodesic.
Study of magnetic geodesics on Heisenberg nilmanifolds.
problem Existence and properties of closed magnetic geodesics on Heisenberg nilmanifolds.
method Analyzing conditions for the existence of closed magnetic geodesics on compact quotients of Heisenberg nilmanifolds.
result Existence of contractible closed magnetic geodesics for any energy level below the Mañé critical value.
The paper explores non-metrizability of projective deformations of Finsler sprays.
problem Investigating non-metrizability of projective deformations of Finsler sprays.
method Analyzing projective deformation of Finsler sprays by holonomy invariant functions and proving non-metrizability for most cases.
result For most values of λ and holonomy invariant nontrivial functions P, the projective deformation is not Finsler metrizable.
New surfaces with special geodesic and horocycle behaviors discovered.
problem Understanding geodesic and horocycle dynamics on hyperbolic surfaces.
method Constructing geometrically infinite hyperbolic surfaces with tailored recurrence properties.
result First examples of non-trivial minimal horocyclic orbit closures and infinite locally-finite conservative horocyclic invariant measures.
We give a complete list of those left invariant unit vector fields on three-dimensional Lie groups with the left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group with the Sasaki metric. As a result, each class of three-dimensional Lie groups admits the totally geodesic…
FPP preserves sublinear Morse boundaries in geodesic graphs.
problem Preserving sublinear Morse boundaries in FPP.
method First passage percolation on geodesic graphs with i.i.d. passage times.
result Sublinear Morse boundaries are invariant under FPP.
The paper examines geometric invariants near a specific type of singular point.
problem The behavior of geometric invariants near a singular point of a surface or curve.
method Analysis of geometric invariants for surfaces and curves that are suspensions of singular curves.
result Evaluation of the orders of Gaussian and mean curvatures for the studied surfaces and curves.
Study geodesic curves on Heisenberg group, classify them, and compute first step of quadrature.
problem Classifying geodesic curves on the Heisenberg group.
method Completely integrable Hamiltonian system, classification of geodesic curves.
result Complete classification of geodesic curves on the Heisenberg group.
We consider equitorsion second type almost geodesic mappings of a non-symmetric affine connection space in this article. Using different computational methods, we obtained some invariants of these mappings. Last generalized Thomas projective parameter and Weyl projective tensor as invariants of a second type almost geo…
We construct differential invariants that vanish if and only if the geodesic flow of a 2-dimensional metric admits an integral of 3rd degree in momenta with a given Birkhoff-Kolokoltsov 3-codifferential.
Paper proves geodesics and focal points unchanged by conformal changes in pseudo-Finsler manifolds.
problem Understanding the behavior of lightlike geodesics in pseudo-Finsler manifolds.
method Used Chern connection, anisotropic calculus, and critical points of energy functional to prove invariance.
result Lightlike geodesics and focal points are preserved by anisotropic conformal changes.
We bring together those systems of hydrodynamical type that can be written as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups with right invariant L2 or H1 metrics. We present their formal derivation starting from Euler's equation, the first order equation satisfied by the ri…
Proves lower bounds on Hausdorff dimension of projections of invariant sets.
problem Lower bounds on Hausdorff dimension of projections of invariant sets.
method Transversal property of geodesics, (k+1)-linear curved Kakeya estimate, Bourgain-Guth argument. result Proves a lower bound on the Hausdorff dimension of projections of invariant sets.
Classifies geodesic flows on projective plane with potential field.
problem Classifying geodesic flows on a projective plane with a potential field.
method Liouville classification and calculation of Fomenko--Zieschang invariants.
result All Fomenko--Zieschang invariants of the system are calculated.
This article considers inverse problems on closed Riemannian surfaces whose geodesic flow is Anosov. We prove spectral rigidity for any Anosov surface and injectivity of the geodesic ray transform on solenoidal 2-tensors. We also establish surjectivity results for the adjoint of the geodesic ray transform on solenoidal…
The Virasoro-Bott group endowed with the right-invariant L2-metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing geodesic distance.