arXiv research

Study shows magnetic trajectories in Berger spheres are homogeneous.

Homogeneous magnetic trajectories in a special linear group proven.

Billiard trajectories and geodesics are closely related geometrically.

New proof shows nonholonomic motions are geodesics, minimizing distance.

Developed a random walk analog of geodesic flow on hyperbolic groups.

Characterizes metabelian distributions and geodesics in sub-Riemannian manifolds.

Paper proves trajectories of Chaplygin systems are reparametrized geodesics.

Connections between Lie derivatives and the deviation equation has been investigated in spaces with affine connection. The deviation equations of the geodesics as well as deviation equations of non-geodesics trajectories have been obtained on this base. This is done via imposing certain conditions on the Lie derivative…

We prove that there exists a geodesic trajectory on the dodecahedron from a vertex to itself that does not pass through any other vertex.

Suppose $(X,J,ω)$ is a Fano manifold and $t \to r_t$ is a diverging Kähler-Ricci trajectory. We construct a bounded geodesic ray $t \to u_t$ weakly asymptotic to $t \to r_t$, along which Ding's $\mathcal F$-functional decreases, partially confirming a folklore conjecture. In absence of non-trivial holomorphic vector fi…

Billiard trajectories in curved spaces have predictable travel times.

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…

Motivated by various results on homogeneous geodesics of Riemannian spaces, we study homogeneous trajectories, i.e. trajectories which are orbits of a one-parameter symmetry group, of Lagrangian and Hamiltonian systems. We present criteria under which an orbit of a one-parameter subgroup of a symmetry group G is a solu…

Study of magnetic geodesics on Heisenberg nilmanifolds.

New constructions show stable geodesics and figure-eights in convex hypersurfaces.

Study integrability of geodesic flow on specific Lie groups.

This study explores magnetic trajectories on the Heisenberg group, finding symmetries and solutions.

New approach constructs symplectic structure on pseudo-Riemannian geodesics.

Study geodesic flows on cones over Riemannian manifolds, showing superintegrability.

We study geodesics along a noncompact Kerr-Newman instanton, where the asymptotic geometry is either de Sitter or anti-de Sitter. We use first integrals for the Hamilton-Jacobi equation to characterize trajectories both near and away from horizons. We study the interaction of geodesics with special features of the metr…

We use the heat flow on the loop space of a closed Riemannian manifold to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined in the spirit of Floer theory by counting, modulo time shift, heat flow trajectories that converge asymptotically…

MFM improves generative model interpolations by learning approximate geodesics on data manifolds.

Magnetic geodesics describe the trajectory of a particle in a Riemannian manifold under the influence of an external magnetic field. In this article, we use the heat flow method to derive existence results for such curves. We first establish subconvergence of this flow to a magnetic geodesic under certain boundedness a…

Superdense flows on surfaces imply bounded geodesics, and vice versa.

Projective geodesic extensions for nonholonomic systems are derived under conformal transformations.

Geodesic extensions for systems with nonholonomic constraints.

Bicycle paths form geodesics in 3D subspaces, related to Kirchhoff rods.

Study particle dynamics in non-differentiable fractal spaces.

New findings on magnetic geodesic flows and periodic motions.

In this paper, we study Bertrand surface offsets by considering the dual geodesic trihedron(dual Darboux frame) of the ruled surfaces. We obtain the relationships between the invariants of Bertrand trajectory ruled surfaces. Furthermore, we obtain the conditions for these surface offset to be developable.

Given a domain or, more generally, a Riemannian manifold with boundary, a billiard is the motion of a particle when the field of force is absent. Trajectories of such a motion are geodesics inside the domain; and the particle reflects from the boundary making the angle of incidence equal the angle of reflection. The bi…

Uniform hyperbolicity is a strong chaotic property which holds, in particular, for Sinai billiards. In this paper, we consider the case of a nonflat billiard, that is, a Riemannian manifold with boundary. Each trajectory follows the geodesic flow in the interior of the billiard, and bounces when it meets the boundary. …

In this paper we analyze the problem of the geodesic connectedness of subsets of Riemannian manifolds. By using variational methods, the geodesic connectedness of open domains (whose boundaries can be not differentiable and not convex) of a smooth Riemannian manifold is proved. In some cases also the convexity of the d…

We investigate the concept of projective equivalence of connections in supergeometry. To this aim, we propose a definition for (super) geodesics on a supermanifold in which, as in the classical case, they are the projections of the integral curves of a vector field on the tangent bundle: the geodesic vector field assoc…

EntroPath learns manifold geometry from diffusion paths.

The paper proves a nonholonomic version of Maupertuis-Jacobi principle and shows that nonholonomic trajectories minimize length.

GAGA learns a warped metric for geometry-aware data generation and interpolation.

Hamilton flows on Kähler manifold for which all trajectories are $H$-planar curves (complex analog of geodesics) are considered. These flows are called $H$-planar. The equation which has to obey the Hamiltonian of $H$-planar Hamilton flow is received and the method of finding general solution of this equation is propos…

Cube edges curves minimize systole length.

Horizon saddle connections imply dense hyperbolic geodesics on dilation surfaces.

Neural solver computes Wasserstein geodesics and velocity fields efficiently.

The paper studies bifurcations in Lagrangian systems and geodesics.

Perturbed geodesics are trajectories of particles moving on a semi-Riemannian manifold in the presence of a potential. Our purpose here is to extend to perturbed geodesics on semi-Riemannian manifolds the well known Morse Index Theorem. When the metric is indefinite, the Morse index of the energy functional becomes inf…

A Monge surface is a surface obtained by sweeping a generating plane curve along a trajectory that is orthogonal to the moving plane containing the curve. Locally, they are characterized as being foliated by a family of planar geodesic lines of curvature. We call surfaces with the latter property PGF surfaces, and inve…

We show that a small perturbation of the boundary distance function of a simple Finsler metric on the $n$-disc is also the boundary distance function of some Finsler metric. (Simple metric form an open class containing all flat metrics.) The lens map is map that sends the exit vector to the entry vector as a geodesic c…

We investigate contact magnetic curves in the real special linear group of degree 2. They are geodesics of the Hopf tubes over the projection curve. We prove that periodic contact magnetic curves in SL(2,R) can be quantized in the set of rational numbers. Finally, we study contact homogeneous magnetic trajectories in S…

Optimal transport learns Riemannian metrics for evolving probability measures.

Study semiclassical measures on complex hyperbolic quotients, identifying measure supports.