Geometry of geodesicsmath.DG
Lecture notes on geodesics in differential geometry.
problem Understanding geodesics in differential geometry.
method Expository lecture notes with exercises.
result Explains the geometry of geodesics.
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A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,695 papers · 148 categories
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48 results for “geodesic geometry”
Classifies geodesic-preserving bijections in Thurston geometries.
problem Identifying bijections that preserve geodesics in different geometries.
method Comprehensive classification and proof for various geometries.
result Complete classification of geodesic-preserving bijections in Thurston geometries.
Geodesic orbit and weakly symmetric properties in spray geometry.
problem Understanding properties of homogeneous spray manifolds.
method Using reductive decompositions and spray vector fields to describe and prove properties.
result Weakly symmetric spray manifolds are geodesic orbit manifolds.
Quantizes geodesics in Kähler and Sasaki geometry.
problem Quantize geodesics in Kähler and Sasaki spaces.
method Classical Fubini-Study map and quantization procedure.
result Proves conditions for geodesics in Kähler potentials.
Paper solves degenerated circle pattern metric problem in spherical geometry.
problem Existence and rigidity of (degenerated) circle pattern metrics with prescribed total geodesic curvatures.
method Defined prescribed combinatorial Ricci flows and studied their convergence.
result First degenerated result for total geodesic curvatures in spherical background geometry.
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.
Geodesic connectedness proved for statistical manifolds with divisible cubic forms.
problem Geodesic connectedness of affine connections on statistical manifolds with divisible cubic forms.
method Analogy with Hopf-Rinow theorem in Riemannian geometry, establishing geodesic completeness.
result Geodesic connectedness established for statistical manifolds with divisible cubic forms.
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.
Recovering manifold geometry from geodesic intersections.
problem Recovering the geometry of a Riemannian manifold from geodesic intersection lengths.
method Applying stitching data to solve the delayed collision data problem.
result Geometry of the manifold can be recovered from geodesic intersection lengths.
Geodesics connect model modes in neural network loss landscapes.
problem Connecting modes in neural network loss landscapes.
method Reframed mode connectivity in Information Geometry, hypothesized geodesics as mode-connecting paths, proposed algorithm to approximate geodesics.
result Geodesics achieve mode connectivity in neural networks.
Geodesics found in deep linear networks.
problem Finding shortest paths in deep neural networks.
method Derived ODEs and explicit solutions for geodesics.
result Horizontal straight lines are geodesics in invariant manifold.
We create real-time geodesic rendering for non-isotropic geometries.
problem Challenging visualization of non-isotropic geometries.
method Novel methods for real-time native geodesic rendering.
result Methods can be applied to visualization, machine learning, and video games.
In this paper, we generalize the classification of geodesic orbit spheres from Riemannian geometry to Finsler geometry. Then we further prove if a geodesic orbit Finsler sphere has constant flag curvature, it must be Randers. It provides an alternative proof for the classification of invariant Finsler metrics with $K\e…
We answer to the question whether a system of the 3rd order ODEs describes geodesics of a conformal structure. We construct a functor from a category of conformal geometries to a category of Cartan geometries associated to the 3rd order ODEs systems. Explicit formulas which define the family of all equations on conform…
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.
New metrics derived from geodesics simplify semi-Riemannian geometry.
problem Simplifying complex semi-Riemannian metrics.
method Constructing plane wave limits along geodesics.
result Generalizes Penrose's limit and encodes tensorial geometry.
We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil-Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equ…
The Sagnac effect is re-examined using Finslerian geometry.
problem Understanding the Sagnac effect in general relativity.
method Reviewing the geometry of the Sagnac effect with a focus on Finslerian metrics.
result An asymmetry in Finslerian metrics affects the Sagnac effect for both future-pointing null and timelike geodesics.
Develops a method to define and characterize geodesics on hyperbolic surfaces.
problem Characterizing closed geodesics on hyperbolic surfaces without self-intersection.
method Constructive definition of the Goldman bracket using closed geodesics.
result Algebraic characterization of geodesics on hyperbolic surfaces.
Symmetric Poisson structures linked to geodesic foliations and Jordan algebras.
problem Understanding geometric structures related to geodesic foliations and dynamics.
method Introducing symmetric Poisson structures, proving correspondences with geodesic foliations and Jordan algebras.
result Symmetric Poisson structures correspond to totally geodesic foliations and Jacobi-Jordan algebras.
The study examines Fisher-Riemann geodesics for nonparametric probability densities.
problem Understanding nonparametric probability densities using Fisher-Riemann geometry.
method Obtaining Fisher-Riemann geodesics as a limit of parametric cases with increasing parameters.
result The weak limit approach for nonparametric probability densities.
Study explores warped geometries of tensor manifolds, finding non-geodesic connections for some parameters.
problem Investigate non-geodesic connections in warped Segre-Veronese manifolds.
method Investigate a one-parameter family of warped geometries, presenting closed expressions for maps and distance.
result Segre-Veronese manifolds are not geodesically connected in Euclidean geometry but can be for some warping parameters.
Developed a new formalism to describe Riemannian geometries using geodesic flow bundles.
problem Understanding the consequences of Einstein equations without solving metric equations.
method Using the bundle of arclength parametrized geodesics (geodesic flow bundle GFB) to describe Riemannian geometry.
result Generalized the cosine- and sine-laws for constant curvature to varying curvature fields.
Study on cut locus of submanifolds in Finsler geometry.
problem Characterizing the cut locus of submanifolds in Finsler manifolds.
method Deformation and characterization of the cut locus, extending previous results.
result Generalization of Klingenberg's lemma for -geodesic loops in reversible Finsler setting.
We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature Kähler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays, and the uniform convexity properties of the space of Kähler metrics.
For non-compact manifolds with boundary we prove that bounded geometry defined by coordinate-free curvature bounds is equivalent to bounded geometry defined using bounds on the metric tensor in geodesic coordinates. We produce a nice atlas with subordinate partition of unity on manifolds with boundary of bounded geomet…
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.
This paper classifies geodesics of projectively flat sprays and introduces a method to determine sprays based on geodesics.
problem Classifying geodesics of projectively flat sprays and determining sprays based on geodesics.
method Introduction of a geodesic method to determine an n-dimensional spray based on a family of curves with 2(n-1) free parameters as geodesics.
result Classification of geodesics of projectively flat sprays and determination of sprays based on geodesics.
A Carter like constant for the geodesic motion in the 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…
We investigate (local) automorphisms of parabolic geometries that generalize geodesic symmetries. We show that many types of parabolic geometries admit at most one generalized geodesic symmetry at a point with non-zero harmonic curvature. Moreover, we show that if there is exactly one symmetry at each point, then the p…
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 Killing magnetic curves in Heisenberg group geometry.
problem Understanding Killing magnetic curves in Heisenberg group.
method Presentation of Heisenberg group geometry and geodesics, study of Killing magnetic curves with explicit formulas.
result Explicit formulas for Killing magnetic curves in Heisenberg group.
New method finds closed timelike geodesics on Lorentzian manifolds.
problem Existence of closed timelike geodesics in Lorentzian geometry.
method Introducing timelike geodesic homotopy and combining with a local length argument.
result Provides new results on the existence of closed timelike geodesics.
A 2008 general overview on Weil-Petersson geometry is offered. A preliminary plan for the subsequent CBMS lectures at Central Connecticut State University is included. Mirzakhani's solution of Witten-Kontsevich is not included - this work essentially requires its own lectures. Lectures on Mirzakhani's Witten-Kontsevich…
The paper explores geometry of probability measures and barycenter maps.
problem Understanding the space of probability measures and their barycenter.
method Information geometry, Fisher metric, dualistic structures, divergences, geodesics.
result Recent developments in the geometry of probability measures and barycenter.
Combinatorial p-th Calabi Flows for Total Geodesic Curvatures in hyperbolic background geometrymath.DG
The paper finds circle packings with specific curvatures in hyperbolic geometry.
problem Finding circle packings with prescribed total geodesic curvatures and discrete Gaussian curvatures.
method Established existence and rigidity via variational principle, introduced combinatorial p-th Calabi flows.
result Introduced combinatorial p-th Calabi flows to find circle packings with prescribed curvatures.
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.
This paper develops GPCA for probability distributions using Otto-Wasserstein geometry.
problem Analyzing modes of variation in datasets of probability measures.
method Geodesic Principal Component Analysis (GPCA) on Wasserstein space with neural networks.
result Identification of geodesic curves that capture modes of variation in probability distributions.
Geodesics found in a metric space of m-subharmonic functions.
problem Metric structure on energy class of m-subharmonic functions.
method Inspired by Kähler geometry, introduced a metric structure and studied metric convergence.
result Geodesics constructed in a subspace of the complete metric space.
Study confirms a 2-sphere metric with three geodesics of minimal length.
problem Understanding the systolic, width, and Gromov-Guth metrics on a 2-sphere.
method Classical min-max and hyperbolic geometry tools.
result Figure-eight geodesics achieve the systolic, width, and Gromov-Guth metrics on a 2-sphere.
Study on curves around a Whitney umbrella focusing on geodesic and normal curvatures.
problem Analyzing geometric properties of curves around a specific surface.
method Examined geodesic and normal curvatures, ruled surfaces, and normal developable surfaces.
result Obtained functions representing geometry on a Whitney umbrella.
There are many equivalent definitions of Riemannian geodesics. They are naturally generalised to sub-Riemannian manifold, but become non-equivalent. We give a review of different definitions of geodesics of a sub-Riemannian manifold and interrelation between them. We recall three variational definitions of geodesics as…
An associated bundle approach to the Bures--Wasserstein geometry of fixed rank covariance matricesmath.DG
Paper explores geometry of covariance matrices using associated bundles.
problem Geometry of fixed-rank covariance matrices.
method Associated bundle approach to Bures--Wasserstein geometry.
result Established a one-to-one correspondence between geodesics.
A new geometry for comparing signals, overcoming traditional limitations.
problem Comparing and interpolating discontinuous and signed signals.
method Investigation of Riemannian geometry on signal space, introducing a metric that measures both horizontal and vertical deformations.
result Characterization of metric properties and establishment of geodesic regularity and stability.
Develops a lifting theory for exponential maps in semi-Riemannian geometry.
problem Overcoming singularities in exponential maps to prove geodesic connectivity.
method Lifting theory for semi-Riemannian manifolds with path-continuation property.
result General path-lifting theorem extending globally under certain conditions.
3D hyperbolic spaces have endless simple paths.
problem Finding simple paths in complex 3D spaces.
method Analyzing geodesics in hyperbolic 3-manifolds.
result Cusped hyperbolic 3-manifolds have infinitely many simple closed geodesics.
Study on the geometry of spacelike hypersurfaces in spacetime.
problem Understanding the geometry of compact spacelike Cauchy hypersurfaces.
method Analysis of a weak Riemannian metric on the manifold of hypersurfaces.
result Positive geodesic distance and non-positive sectional curvature.
We study the geometry of the Thurston metric on Teichmuller space by examining its geodesics and comparing them to Teichmuller geodesics. We show that, similar to a Teichmuller geodesic, the shadow of a Thurston geodesic to the curve graph is a reparametrized quasi-geodesic. However, we show that the set of short curve…