Geodesic walks converge to Brownian motion on Finsler manifolds.
problem Understanding random walks on Finsler manifolds.
method Analyzing convergence of geodesic random walks to diffusion processes.
result The Brownian motion on a Riemannian metric is a key result.
Geodesics in R^n configuration spaces for points apart by epsilon.
problem Finding paths between points in R^n with minimum distance constraints.
method Explicit formulas for geodesics in configuration spaces with Euclidean metric.
result Geodesic motion-planning rules for configuration spaces of ordered pairs of points.
New findings on magnetic geodesic flows and periodic motions.
problem Characterizing superintegrable systems in magnetic geodesic flows.
method Analyzing rotationally symmetric magnetic geodesic flows.
result All sufficiently slow motions in a central magnetic field are periodic under specific curvature and homogeneity conditions.
Study random walks on sub-Riemannian manifolds using retractions.
problem Modeling random walks on sub-Riemannian manifolds.
method Use retractions to approximate normal geodesics and study convergence to Brownian motion.
result Convergence of geodesic random walks defined with different connections.
Study of geodesics on SL(n) with Hilbert-Schmidt metric, revealing complex dynamics in higher dimensions.
problem Geodesics on SL(n) with Hilbert-Schmidt metric.
method Analysis of geodesics, use of Virial-identity-based criterion, study of explicit families of solutions, classification of geodesics.
result Complex dynamics in higher dimensions, existence of bounded geodesic motions in even dimensions, instability of swirling and shear flows in even dimensions.
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.
Study geodesic complexity in homogeneous Riemannian manifolds.
problem Geodesic motion planning and complexity in homogeneous Riemannian manifolds.
method Riemannian geometry, stratifications of cut loci, and properties of homogeneous manifolds.
result Established new bounds on geodesic complexity and computed its value for homogeneous Riemannian manifolds.
Study on Brownian motion on discrete curve spaces, proving stochastic completeness.
problem Analyzing Brownian motion on spaces of discrete curves.
method Introduced and studied Brownian motion on spaces of discrete regular curves with Sobolev-type metrics.
result All geodesically complete spaces of discrete regular curves are stochastically complete.
Unified framework for human motion generation on Riemannian manifolds.
problem Learning valid human motion in Euclidean spaces.
method Riemannian Motion Generation (RMG) on product manifolds, Riemannian flow matching.
result Achieves state-of-the-art FID (0.043) on HumanML3D and surpasses strong baselines on MotionMillion.
This paper classifies Legendre singularities of sub-Riemannian geodesics on surfaces.
problem Classifying singularities of sub-Riemannian geodesics.
method Complete local classification using Legendre fibrations.
result Legendre singularities are completely classified for sub-Riemannian geodesics.
Study nonrigid dynamics of unitary groups on Lie groups via kinetic energy metrics.
problem Understanding the dynamics of unitary groups on Lie groups using kinetic energy metrics.
method Least action principle applied to geodesics of the kinetic energy metric on G. result Kinetic energy metric on G is not complete and not invariant. New cutoff phenomenon found for geodesic paths on hyperbolic manifolds.
problem Understanding the cutoff phenomenon for geodesic paths on hyperbolic manifolds.
method Spectral strategy and detailed spectral analysis of the spherical mean operator.
result Geodesic paths on compact hyperbolic manifolds exhibit cutoff for spatially localized initial conditions.
Classifies 3-braids from choreographic motions on Lissajous curves, linking them to mapping classes and geodesics.
problem Classifying 3-braids from choreographic motions on Lissajous curves.
method Parametrization in terms of levels and slopes, using dilatation and geodesic cutting sequences.
result Dilatation of pseudo-Anosov mapping classes increases with level or slope.
Proves CLT for Brownian paths on pinched negative curvature manifolds.
problem Distribution of Brownian paths on pinched negative curvature manifolds.
method Proof of central limit theorem for distances and Green functions.
result Central limit theorem holds for Brownian paths in pinched negative curvature.
We continue our research work started in "Kinematic Quantities and Raychaudhuri Equations in a 5D Universe" (Eur. Phys. J. C, 2015), and obtain in a covariant form, the equations of motion with respect to the (1+1+3) threading of a 5D universe (Mˉ,gˉ). The natural splitting of the tangent bundle of $…
A Carter like constant for the geodesic motion in the Y(p,q) Einstein-Sasaki geometries is presented. This constant is functionally independent with respect to the five known constants for the geometry. Since the geometry is five dimensional and the number of independent constants of motion is at least six, the geode…
The paper studies how test particles' mass and charge vary in Kaluza-Klein models.
problem Understanding how test particles' mass and charge change in Kaluza-Klein models.
method Analyzes geodesic motion in a 5D Kaluza-Klein spacetime with background metrics encoding 4D gauge fields and Higgs-like scalars.
result The mass and charge of test particles become variable when traversing regions with massive gauge fields or non-constant Higgs scalars.
New model for visual cortex border completion using bicycle wheel motions.
problem Understanding border completion in the visual cortex V1.
method Sub-Riemannian Hamiltonian formalism and bicycle wheel analogy.
result Analogies between visual cortex border completion and bicycle wheel motions.
New connections share geodesics with superintegrable systems.
problem Understanding geodesics in affine connections related to superintegrable systems.
method Analyzing dual-geodesics and comparing them across different connections.
result Certain torsion-free affine connections associated with second order superintegrable systems share the same dual-geodesics.
Simple geodesics on spherical tetrahedra identified for specific angles.
problem Identifying simple closed geodesics on regular tetrahedra in spherical space.
method Analyzing pairs of coprime integers (p,q) to find angles α1 and α2.
result Existence and non-existence of simple closed geodesics for specific angles.
Survey on manifold complexities and motion planning in robotics.
problem Understanding topological complexities of manifolds in robotic motion planning.
method Overview of topological complexities, geodesic motion planning, and connections to critical point theory.
result Estimation of motion planning complexity using Riemannian geometry and critical point theory.
New proof shows nonholonomic motions are geodesics, minimizing distance.
problem Nonholonomic motion equations are not variational.
method Proved geodesic property of nonholonomic trajectories using Riemannian metrics.
result Nonholonomic motions minimize distance in their manifold.
In this article, we will formulate a mathematical framework that allows us to treat character animations as points on infinite dimensional Hilbert manifolds. Constructing geodesic paths between animations on those manifolds allows us to derive a distance function to measure similarities of different motions. This appro…
Classifies surfaces with zero mean curvature in a light cone.
problem Classifying surfaces with zero mean curvature in a light cone.
method Examined geodesics and screw motions, used Weierstrass representations.
result Complete classification of ruled zero mean curvature surfaces.
In considering the mathematical problem of describing the geodesics on a torus or any other surface of revolution, there is a tremendous advantage in conceptual understanding that derives from taking the point of view of a physicist by interpreting parametrized geodesics as the paths traced out in time by the motion of…
Study of motion control systems on Lie groups with specific geometric constraints.
problem Controlling motion systems on Lie groups with geometric constraints.
method Analysis of control systems on Lie groups, focusing on infinitesimal roto-translations and geodesics.
result Explicit geodesics found for the sub-Riemannian structure on the Lie group.
We expose some ideas from mathematical logics, i.e. the background of the theory of o-minimal structures, and demonstrate how they lead to the notion of a tame integral of motion and some extensions and clarifications of previous results on obstructions to integrability of geodesic flows.
We study the geometry of the inextensible string (the whip) and its discrete approximation (the chain). In the absence of gravity, both motions represent geodesic motions on certain manifolds. We show how the motion of the chain converges to that of a whip, and how the curvature of the chain's configuration space conve…
In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean C2-smooth surface in the affine group and the group of rigid motions of the Minkowski plane away from characteristic points and signed geodesic curvature for Euclidean C2-smooth curves on surfaces. We get Gauss-Bonnet theorems i…
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…
New metric invariant GC connects to TC, with applications in robotics.
problem Understanding motion planning in metric spaces.
method Introducing geodesic complexity (GC) as a new invariant.
result GC and topological complexity (TC) coincide in many cases but can be distinguished.
We establish an integral test describing the exact cut-off between recurrence and transience for normally reflected Brownian motion in certain unbounded domains in a class of warped product manifolds. Besides extending a previous result by R. Pinsky, who treated the case in which the ambient space is flat, our result r…
Study helical motions of lines in 3D spaces, solving control problems.
problem Controlling helical motions of lines in 3D spaces.
method Analyzing control systems on manifolds of oriented geodesics in 3D spaces of different curvatures.
result The system is controllable if and only if alpha^2 ≠ kappa.
Geodesics spiral around Reeb orbits in 3D contact manifolds.
problem Understanding geodesics in sub-Riemannian geometry.
method Normal form along Reeb orbits due to Melrose.
result Sub-Riemannian geodesics spiral around Reeb orbits in both phase and configuration spaces.
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 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…
The paper derives Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.
problem Computing curvature and geodesic curvature for surfaces and curves in affine and rigid motions groups.
method Defined deformed Schouten-Van Kampen connections, computed Gaussian curvature limits, and signed geodesic curvature.
result Derived Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.
We develop a Malliavin calculus on the horizontal path space of a totally geodesic Riemannian foliation. As a first application, under suitable assumptions, we prove a log-Sobolev inequality for a natural one-parameter family of infinite-dimensional Ornstein-Uhlenbeck type operators. As a second application, we obtain …
Study radial processes in sub-Riemannian Brownian motions, proving stochastic completeness and eigenvalue estimates.
problem Analyzing sub-Riemannian Brownian motions and their radial processes.
method Application of Itô's formula and sub-Laplacian comparison theorems to prove stochastic completeness and eigenvalue estimates.
result Proved Cheng's type estimates for Dirichlet eigenvalues of sub-Riemannian metric balls.
We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global behavior may be periodic or the curve may be dense in a Clifford torus embedded…
The equations of motion of a charged ideal fluid, respectively the superconductivity equation (both in a given magnetic field) are showed to be geodesic equations on a general, respectively central extension of the group of volume preserving diffeomorphisms with right invariant metric. For this, quantization of the mag…
The paper studies symmetry reduction and optimal control on Riemannian manifolds.
problem Symmetry reduction and optimal control on Riemannian manifolds.
method Derivation of reduced equations of motion for variational problems on Lie groups and Riemannian homogeneous spaces.
result Derivation of geodesic equations and reduced equations of motion for various applications.
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.
Geodesic extensions for systems with nonholonomic constraints.
problem Extending equations of motion for systems with nonholonomic constraints.
method Constructing extensions to second-order ODEs, investigating geodesic conditions.
result Conditions for nonholonomic trajectories to be geodesics of a Riemannian metric.
In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus T2 for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic…
Unified framework for Brownian motion distances on specific geometric manifolds.
problem Understanding Brownian motion distances on radially isoparametric manifolds.
method Developed a geometric framework and derived drift-window inequalities.
result Unified framework for coadapted Brownian couplings on RIM.
We consider the motion of small bodies in general relativity. The key result captures a sense in which such bodies follow timelike geodesics (or, in the case of charged bodies, Lorentz-force curves). This result clarifies the relationship between approaches that model such bodies as distributions supported on a curve, …
We consider the four-dimensional nonholonomic distribution defined by the 4-potential of the electromagnetic field on the manifold. This distribution has a metric tensor with the Lorentzian signature (+,−,−,−), therefore, the causal structure appears as in the general relativity theory. By means of the Pontryagin's m…