Study on scattering geodesics on modular surface and their sojourn times.
problem Distribution of scattering geodesics and their sojourn times on modular surface.
method Analysis of scattering geodesics in modular surface, establishing connection to prime divisors in arithmetic progression.
result Established a connection between scattering geodesics and prime divisors in arithmetic progression.
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.
Study geodesics on Grushin spaces, proving upper bounds on conjugate times.
problem Classify geodesics on higher-dimensional Grushin spaces.
method Solve Hamilton's equations using calculus of generalized trigonometric functions, analyze symmetries, and use density arguments.
result Prove a conjectured cut time provides an upper bound on conjugate times.
Study geodesics entering a fixed cusp neighborhood multiple times.
problem Understanding geodesics entering a specific cusp neighborhood multiple times.
method Investigate reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
result Characterized the class of reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
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.
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 SU(1,1)imesR and SO0(2,1)imesR. result Found geodesics, shortest arcs, cut loci, and conjugate loci.
This paper explores the relation between convex functions and the geometry of space-times and semi-Riemannian manifolds (an investigation initiated by Gibbons-Ishibashi). Specifically, we study geodesic connectedness. We give geometric-topological proofs of geodesic connectedness for classes of space-times to which kno…
Lightlike hypersurfaces in cone structures minimize time.
problem Finding time-minimizing paths in cone structures.
method Defining lightlike hypersurfaces and proving their foliation by cone geodesics.
result Lightlike hypersurfaces in globally hyperbolic spacetimes are time-minimizing.
Study geodesics in Kähler metrics for all time.
problem Understanding geodesics in Kähler metrics for all time.
method Analyzing geodesics as induced by holomorphic vector fields.
result Derivative of geodesics implies a variant of convexity theorem.
Study geodesic properties of time series data using Wasserstein metric.
problem Modeling nonlinear time series with transport-based metrics.
method Generalized Wasserstein metric and signed cumulative distribution transforms.
result Geodesic properties provide added interpretability and robustness in time series classifiers.
Shortest geodesic on curved spheres is no longer than 3 times the diameter.
problem Finding the shortest closed geodesic on spheres with positive curvature.
method Proved a new isoperimetric inequality for spheres with pinched curvature, used to improve the bound on the shortest geodesic.
result The shortest closed geodesic is no longer than 3 times the diameter of the sphere.
Let M be a Margulis spacetime whose associated complete hyperbolic surface S has compact convex core. Generalizing the correspondence between closed geodesics on M and closed geodesics on S, we establish an orbit equivalence between recurrent spacelike geodesics on M and recurrent geodesics on S. In contrast, no timeli…
Geodesics of contactomorphisms on a specific manifold are characterized by Hamiltonian functions.
problem Characterizing geodesics of contactomorphisms on a manifold with a standard contact structure.
method Analyzing geodesics defined by different norms on the identity component of the group of contactomorphisms.
result The norm of a geodesic contactomorphism can be expressed in terms of the maximum of the Hamiltonian function.
On a hyperbolic Riemann surface, given two simple closed geodesics that intersect n times, we address the question of a sharp lower bound Ln on the length attained by the longest of the two geodesics. We show the existence of a surface Sn on which there exists two simple closed geodesics of length Ln interse…
Study geodesic orbit metrics in quaternionic Stiefel manifolds.
problem Characterize geodesic orbit spaces in quaternionic Stiefel manifolds.
method Analyze homogeneous Riemannian spaces (M=G/H,g) with geodesics as orbits of subgroups. result Identify conditions for $(\Sp(n)/\Sp(n_1) imes \cdots imes \Sp(n_s), g)$ to be a geodesic orbit space.
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.
Study geodesics on compact Lorentzian solvmanifolds, finding conditions for closedness.
problem Conditions for closed geodesics on compact Lorentzian solvmanifolds.
method Analyzing geodesics on Lorentzian homogeneous spaces of solvable Lie groups.
result Conditions for every lightlike geodesic to be closed on quotient spaces.
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.
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…
Totally geodesic Lagrangian submanifolds in nearly Kähler S³×S³.
problem Characterizing Lagrangian submanifolds in nearly Kähler manifolds.
method Analyzing H-umbilical properties and their implications for geodesicity. result In nearly Kähler S³×S³, H-umbilical Lagrangian submanifolds are totally geodesic. Researchers found sub-Lorentzian geodesics on a specific Lie subgroup.
problem Finding geodesics on a specific Lie subgroup with a sub-Lorentzian metric.
method Formulated a time-anti-optimal control problem, applied Pontryagin's minimum principle, and used geodesics and shortest arcs of a sub-Riemannian metric.
result Discovered sub-Lorentzian nonspacelike geodesics and longest arcs.
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.
Study of Lévy flights on Zoll surfaces, revealing geometric information.
problem Understanding the mean first capture time of Lévy flights on Zoll surfaces.
method Analysis of geodesic Lévy processes on Zoll surfaces, focusing on the first correction term.
result The first correction term encodes geometric information, specifically the degree of the conjugate point.
We study homologically maximizing timelike geodesics in conformally flat tori. A causal geodesic γ in such a torus is said to be homologically maximizing if one (hence every) lift of γ to the universal cover is arclength maximizing. First we prove a compactness result for homologically maximizing timelike geodesics…
Totally geodesic submanifolds in product spaces imply special curvature properties.
problem Characterizing manifolds based on the existence of totally geodesic submanifolds.
method Analyzing totally geodesic submanifolds in products of non-positively curved manifolds with transversality conditions.
result If infinitely or just a single dense such submanifolds exist, the manifold must be locally symmetric.
Study of time-dependent metrics and connections in geometry.
problem Understanding geodesics and connections in time-dependent Riemannian manifolds.
method Examine connections on product manifolds, explore parallel transport, geodesics, and torsion.
result Define the derivative of a one-parameter family of connections.
Study classifies totally geodesic surfaces in nearly Kähler space.
problem Classifying totally geodesic surfaces in nearly Kähler space.
method Detailed description of nearly Kähler space and classification of surfaces.
result Classification of totally geodesic almost complex surfaces.
An affine manifold is said to be geodesically complete if all affine geodesics extend for all time. It is said to be affine Killing complete if the integral curves for any affine Killing vector field extend for all time. We use the solution space of the quasi-Einstein equation to examine these concepts in the setting o…
We prove Wadsley's theorem for foliations by closed non-lightlike geodesics. As an application we show that every pseudo-Riemannian and non-Riemannian 2-mainfold, all of whose time- or spacelike geodesics are closed, is diffeomorphic to S1×R. Further we show that every pseudo-Riemannian 2-manifold with index …
This study examines abnormal geodesics in 2D-Zermelo navigation problems, revealing their role in separating time minimal and maximal curves.
problem The role of abnormal geodesics in planar Zermelo navigation problems with strong current.
method Geometric time optimal control approach, focusing on the heading angle of the ship.
result Abnormal geodesics separate time minimal and maximal curves, and are both small-time minimizing and maximizing.
Geodesics on hyperbolic surfaces become evenly spread over time.
problem Distribution of geodesics on hyperbolic surfaces.
method Equidistribution analysis of geodesics with length less than T as T approaches infinity.
result Closed geodesics on hyperbolic surfaces of finite area become evenly spread over time.
Paper proves a new criterion for time-like geodesics in flat spacetimes.
problem Existence and nature of time-like geodesics in asymptotically flat spacetimes.
method Generalized topological criterion using the Jordan-Brouwer Separation Theorem and differential geometry.
result Conclusively affirms the presence of time-like geodesics intersecting transversally.
This survey explores compact geodesic orbit manifolds and their properties.
problem Classifying compact geodesic orbit manifolds.
method Review and analysis of existing results.
result Study of geodesic orbit condition for $\SU(5)/\s(\U(2) imes \U(2))$.
Study geodesic orbit metrics on specific homogeneous spaces.
problem Characterize geodesic orbit spaces in a class of homogeneous bundles.
method Analyze geodesic orbit spaces of compact Lie groups with semisimple subgroups.
result Identify conditions for a metric to be geodesic orbit.
Study proves convergence of quantized geodesics to Mabuchi geodesics.
problem Convergence of quantized geodesics to Mabuchi geodesics in short time.
method Real-analytic initial data and convergence proof.
result Proves convergence of quantized Bergman geodesics to Mabuchi geodesics.
In this paper we study 1/k geodesics, those closed geodesics that minimize on all subintervals of length L/k, where L is the length of the geodesic. We develop new techniques to study the minimizing properties of these curves on doubled polygons, and demonstrate a sequence of doubled polygons whose closed geodesics…
We define the notion of geodesic completeness for semi-Riemannian metrics of low regularity in the framework of the geometric theory of generalized functions. We then show completeness of a wide class of impulsive gravitational wave space-times.
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.
The paper proves the existence and properties of geodesics on convex surfaces.
problem Existence and properties of geodesics on convex surfaces with free boundaries.
method Free boundary curve shortening flow on closed surfaces with strictly convex boundary.
result Existence of two free boundary embedded geodesics and geodesics with Morse Index 1 and 2.
The study reveals conditions for infinite closed geodesics on specific surfaces.
problem Conditions for infinite closed geodesics on complete surfaces.
method Analyzes geometric and homological properties of closed geodesics on cylinders and planes.
result Proves that complete cylinders with isolated geodesics have zero, one, or infinitely many homologically visible geodesics.
Some results about the geodesic boundary of minimal surfaces in H2×R are generalized for surfaces of constant mean curvature surfaces H, with 0≤H≤1/2.
Geodesics in curved spaces spread evenly over time.
problem Equidistribution of geodesics in negatively curved spaces.
method Proving equidistribution of geodesic flow orbits towards measures of maximal entropy and Bowen-Margulis measure.
result Equidistribution of divergent geodesics in negative curvature as their complexity increases.
In this paper we describe the stable and unstable leaves for the geodesic flow on the space of non-wandering spacelike geodesics of a Margulis Space Time and prove contraction properties of the leaves under the flow. We also show that monodromy of Margulis Space Times are "Anosov representations in non semi-simple Lie …
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…
Erdős-Kac theorem applied to geodesics on modular surface.
problem Understanding the distribution of geodesics on modular surfaces.
method Analyzing the number of scattering geodesics with a fixed sojourn time.
result Gaussian behavior for the number of scattering geodesics on modular surface.
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.
The paper classifies totally geodesic Lagrangian submanifolds in a specific pseudo-nearly Kähler space.
problem Understanding Lagrangian submanifolds in pseudo-nearly Kähler spaces.
method Examining four classes of submanifolds based on their behavior with respect to an almost product structure, then classifying totally geodesic ones.
result A complete classification of totally geodesic Lagrangian submanifolds in the pseudo-nearly Kähler SL(2,R)imesSL(2,R). In this paper we consider finite volume hyperbolic manifolds X with non-empty totally geodesic boundary. We consider the distribution of the times for the geodesic flow to hit the boundary and derive a formula for the moments of the associated random variable in terms of the orthospectrum. We show that the the first tw…