Conditions for geodesics to project to curves of constant curvature found.
problem Understanding geodesics in sub-Riemannian manifolds.
method Analyzing Riemannian submersions and geodesic curvature properties.
result Necessary and sufficient conditions for geodesics to project to curves of constant curvature.
Researchers define geodesic curvature for 3D sub-Riemannian curves, improving distance calculations.
problem Improving sub-Riemannian distance calculations for 3D curves.
method Introducing geodesic curvature k ζ k_ζ k ζ for smooth horizontal curves in 3D contact sub-Riemannian manifolds. result Geodesic curvature appears as the first corrective term in the Taylor expansion of sub-Riemannian distance.
New method for curvature computation in sub-Riemannian geometry.
problem Computing curvature in sub-Riemannian manifolds.
method Using compatible affine connections and induced tensors.
result Universal Bonnet-Myers theorem for sub-Riemannian geometry.
Study geodesic curvature in Heisenberg group, interpreting it as distance correction.
problem Interpreting geodesic curvature in the Heisenberg group.
method Analyzing smooth horizontal curves in the Heisenberg group, interpreting curvature as distance correction.
result Geodesic curvature in Heisenberg group is the first term in distance expansion.
The Heisenberg group's curvature and Gauss-Bonnet theorem are explored using Riemannian approximation.
problem Defining curvature in the Heisenberg group for smooth surfaces and curves.
method Using a Riemannian approximation scheme to define sub-Riemannian Gaussian and signed geodesic curvatures.
result Proved a Heisenberg version of the Gauss-Bonnet theorem.
Study shows only hyperplanes in Heisenberg groups have zero curvature.
problem Understanding Bernstein problem in higher dimensional Heisenberg groups.
method Sub-Riemannian characterization of ruling property and study of geodesics.
result Only hyperplanes have zero horizontal symmetric second fundamental form in Heisenberg groups.
Study shows how brain completes missing parts of curves and constructs surfaces of negative curvature.
problem How the brain completes missing parts of curves and constructs surfaces of negative curvature.
method Solving variational problems to find sub-Riemannian geodesics and constructing surfaces of constant negative curvature.
result There is a one-to-one correspondence between sub-Riemannian geodesics used by the brain and rotational surfaces of constant negative curvature.
Study on homogeneous geodesics in sub-Riemannian geometry.
problem Characterizing and understanding homogeneous geodesics in sub-Riemannian manifolds.
method Criterion for geodesics to be homogeneous, proof of geodesic orbit spaces, examples of geodesic orbit sub-Riemannian manifolds.
result Existence of at least one homogeneous geodesic under broad conditions.
The study explores different definitions of geodesics in sub-Riemannian geometry.
problem Understanding geodesics in sub-Riemannian geometry and their equivalence.
method Review of three variational definitions of geodesics and three definitions of straightest curves.
result Shortest geodesics coincide with straightest geodesics in some sub-Riemannian manifolds.
Geometric characterization of sub-Riemannian geodesics on frame bundles.
problem Characterize sub-Riemannian geodesics on frame bundles of 3-manifolds.
method Lie theoretical description, geometric characterization, complex length spectrum computation.
result Sub-Riemannian metrics on frame bundles of isospectral manifolds are length isospectral.
The paper introduces a new flow for Legendrian curves in Sasakian sub-Riemannian 3-manifolds.
problem Investigating the behavior of Legendrian curves in specific geometric settings.
method Introducing and analyzing a modified inverse mean curvature flow for Legendrian curves in Sasakian sub-Riemannian 3-manifolds.
result The flow preserves the Legendrian condition and increases the length of curves, with specific asymptotic behaviors.
We prove sectional and Ricci-type comparison theorems for the existence of conjugate points along sub-Riemannian geodesics. In order to do that, we regard sub-Riemannian structures as a special kind of variational problems. In this setting, we identify a class of models, namely linear quadratic optimal control systems,…
Study new volume invariant in Hamiltonian dynamics.
problem Volume behavior in Hamiltonian dynamics.
method Introduce new invariant and study its properties.
result Generalize Riemannian volume expansion to sub-Riemannian manifolds.
Study geodesics in sub-Riemannian manifolds, resolving open questions.
problem Understanding geodesics in sub-Riemannian geometry, especially those that lose regularity.
method Constructing examples and using a lifting procedure.
result Existence of non-smooth and branching minimizing geodesics in real-analytic sub-Riemannian manifolds and Carnot groups.
Study of geodesic branching in 2D sub-Riemannian manifolds.
problem Branching of geodesics in sub-Riemannian manifolds of rank two.
method Analysis of geodesic behavior in sub-Riemannian geometry.
result Continuous families of strictly abnormal branching geodesics and accumulation of branching points.
Study validates Michor-Mumford conjecture in infinite dimensional Hilbertian H-type groups.
problem Validation of Michor-Mumford conjecture in infinite dimensional Hilbertian H-type groups.
method Introduced infinite dimensional Hilbertian H-type groups with weak, graded, left invariant Riemannian metrics and proved the vanishing of geodesic distance and local unboundedness of sectional curvature.
result Validation of Michor-Mumford conjecture linking geodesic distance vanishing to local unboundedness of sectional curvature.
We found all lengths of closed sub-Riemannian paths on a sphere.
problem Finding lengths of closed sub-Riemannian paths on a sphere.
method Elementary methods avoiding explicit formulas.
result Determined all lengths of closed sub-Riemannian geodesics.
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.
In the present paper we show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This helps us to describe the geodesic flow of sub-Riemannian metrics on totally geodesic Riemannian submersions…
Study sub-Riemannian geodesics on a Heisenberg 3D nil-manifold.
problem Characterize geodesics on a specific nil-manifold.
method Analyze the projection of sub-Riemannian structure, describe geodesic flow dynamics, and estimate sub-Riemannian geodesics.
result Sharp bounds and estimates for sub-Riemannian geodesics.
New integrable structures found with abnormal geodesics.
problem Integrable homogeneous sub-Riemannian structures with abnormal geodesics.
method Analysis of equivalence problem for sub-Riemannian Engel structures.
result First known family of examples of integrable homogeneous sub-Riemannian structures with strictly abnormal geodesics.
Study on geodesics in a specific sub-Riemannian structure with two types of behavior.
problem Symmetries and geodesics in a sub-Riemannian structure with growth vector (4,7).
method Symmetry reduction to analyze geodesics.
result Existence of two distinct types of geodesics: those that avoid the fixed point set and those that are contained within it.
Study optimal transport in 4D sub-Riemannian spaces with many singular geodesics.
problem Existence and uniqueness of optimal transport maps in sub-Riemannian structures.
method Analysis of Monge optimal transport problem in sub-Riemannian manifolds.
result Extension of previous results to sub-Riemannian structures of rank two in 4D.
Study sub-Riemannian structures on Banach manifolds, extending controllability and geodesic flow results.
problem Analyzing sub-Riemannian structures on Banach manifolds.
method Define sub-Riemannian structures, extend Chow-Rashevski theorem, and provide conditions for Hamiltonian geodesic flow.
result Extensions of the Chow-Rashevski theorem for exact controllability and conditions for Hamiltonian geodesic flow in infinite-dimensional setting.
We extend Agrachev-Brockett-Jurjdevic's solution to normal sub-Riemannian geodesics.
problem Finding normal sub-Riemannian geodesics in Lie group structures.
method Constructing sub-Riemannian structures from Lie subalgebra filtrations and applying to homogeneous spaces.
result Explicit solutions for normal geodesics in general chains of Lie subgroups.
Derivatives of sub-Riemannian geodesics are always L p L_p L p -Hölder continuous.
problem Smoothness of sub-Riemannian geodesics
method Proving L p L_p L p -Hölder continuity of derivatives result Derivatives of sub-Riemannian geodesics are L p L_p L p -Hölder continuous Researchers find metric lines in SE(2) using Hamilton-Jacobi theory.
problem Identifying metric lines in the Special Euclidean group on the plane.
method Alternative proof using Hamilton-Jacobi theory.
result Metric lines in SE(2) are identified.
The abstract discusses how sub-Riemannian manifolds can have branching geodesics.
problem The existence of branching geodesics in sub-Riemannian geometry.
method Analyzing the rank discontinuity of normal geodesics and constructing specific examples.
result Sub-Riemannian manifolds can contain branching normal minimizing geodesics.
Characterizes metabelian distributions and geodesics in sub-Riemannian manifolds.
problem Characterizing metabelian distributions and geodesics in sub-Riemannian manifolds.
method Characterization of metabelian distributions in terms of principal bundle structures. Proof of geodesic properties for rank-2 distributions.
result For rank-2 metabelian distributions, geodesics are of class C 1 C^1 C 1 . The paper studies geodesics in Carnot groups and their properties.
problem Understanding geodesics in sub-Riemannian and sub-Finsler manifolds.
method Analyzing infinitesimal and asymptotic properties of geodesics in Carnot groups.
result Blowups of geodesics in Carnot groups are still geodesics in lower rank groups.
The present paper is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions. Using Pontryagin Maximum Principle, we treat Riemannian and sub-Riemannian cases in an unified way and obtain some algebraic necessary conditions for the geodes…
The paper calculates curvature limits and proves Gauss-Bonnet theorems in affine and Minkowski groups.
problem Computing curvature limits in affine and Minkowski groups.
method Analyzing Euclidean C 2 C^2 C 2 -smooth surfaces and curves in affine and Minkowski groups. result Gauss-Bonnet theorems in affine and Minkowski groups are proven.
Researchers discovered geodesics and shortest arcs on S L ( 2 ) SL(2) S L ( 2 ) Lie group.
problem Finding shortest paths on a specific Lie group.
method Left-invariant sub-Riemannian metric analysis.
result Geodesics and shortest arcs identified on S L ( 2 ) SL(2) S L ( 2 ) . We consider the problem P c u r v e \mathbf{P_{curve}} P curve of minimizing ∫ 0 L ξ 2 + κ 2 ( s ) d s \int \limits_0^L \sqrt{ξ^2 + κ^2(s)} \, {\rm d}s 0 ∫ L ξ 2 + κ 2 ( s ) d s for a curve x \mathbf{x} x on R \mathbb R R with fixed boundary points and directions. Here the total length L ≥ 0 L\geq 0 L ≥ 0 is free, s s s denotes the arclength parameter, κ κ κ denotes the absolute curvature of $\mathbf{x}…
Study on sub-Riemannian geometry in 4D, focusing on abnormal geodesics.
problem Understanding sub-Riemannian geometry and its spectrum.
method Wave trace expansion, Weyl laws, propagation of singularities, quantum ergodicity.
result First appearance of abnormal geodesics in sub-Riemannian spectral geometry.
Study curvature in Carnot groups with rank-two distributions.
problem Understanding curvature in specific geometric structures.
method Generalized sectional curvature for Carnot groups with rank-two distributions.
result Curvature depends on the Engel part in certain Carnot groups.
Study periodic geodesics on contact 3D manifolds, proving existence and precise properties.
problem Existence and properties of periodic geodesics in contact sub-Riemannian metrics.
method Develops two independent subjects: existence of spiraling geodesics and precise study of geodesics on quotient of SL2(R).
result Proves existence and precise properties of periodic geodesics.
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.
Study shows not all smooth paths are optimal in certain geometric structures.
problem Existence of non-smooth sub-Riemannian minimizing geodesics.
method Constructed a C 2 C^2 C 2 but not C 3 C^3 C 3 length-minimizer example. result Found a real-analytic sub-Riemannian structure with non-smooth minimizers.
Evolution equations for paths on manifolds with anisotropic weights and constraints.
problem Inference on non-trivial covariance manifold-valued data.
method Anisotropically weighted curve energies, sub-Riemannian metrics, geodesics, stochastic development.
result Evolution equations derived for paths on manifolds, connecting to geodesics and polynomials.
The paper calculates curvature limits and Gauss-Bonnet theorems in the Heisenberg group.
problem Computing curvature limits and Gauss-Bonnet theorems in the Heisenberg group.
method Sub-Riemannian limits of Gaussian curvature, Schouten-Van Kampen affine connections, and adapted connections.
result Gauss-Bonnet theorems associated with Schouten-Van Kampen affine connections in the Heisenberg group.
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 on geodesics in Cartan group sub-Riemannian problem, proving conjugate time relation to Maxwell time.
problem Geodesics in sub-Riemannian problem on Cartan group.
method Analysis of symmetries, geodesic optimality, conjugate time calculation.
result First conjugate time is not less than Maxwell time, and equal for certain geodesics.
New optimality conditions for sub-Riemannian geodesics derived.
problem Optimality conditions for sub-Riemannian geodesics.
method Geometric translation and ODE derivation.
result New second-order necessary optimality conditions.
Study geodesics and shortest arcs on Lie groups with specific metrics.
problem Characterize geodesics and shortest paths on Lie groups with sub-Riemannian metrics.
method Analytical and geometric methods to find geodesics and shortest arcs.
result Found geodesics, shortest arcs, distances, and conjugate loci for specified metrics.
Analytic sub-Riemannian geodesics in 3D are always C1 and analytic except finitely many points.
problem Regularity of sub-Riemannian minimizing geodesics in 3D analytic manifolds.
method Investigation of totally nonholonomic analytic distributions, proof of Hausdorff dimension, and regularity of minimizing geodesics.
result Minimizing sub-Riemannian geodesics in 3D analytic manifolds are C1 and analytic except finitely many points.
Study geodesics and shortest arcs on Lie groups with specific metrics.
problem Characterize geodesics and shortest arcs in sub-Riemannian metrics on Lie groups.
method Investigated left-invariant sub-Riemannian metrics on S U ( 1 , 1 ) i m e s R SU(1,1) imes\mathbb{R} S U ( 1 , 1 ) im es R and S O 0 ( 2 , 1 ) i m e s R SO_0(2,1) imes\mathbb{R} S O 0 ( 2 , 1 ) im es R . result Found geodesics, shortest arcs, cut loci, and conjugate loci.
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.