Approximate symmetries of geodesic equations on 2-spheres are studied. These are the symmetries of the perturbed geodesic equations which represent approximate path of a particle rather than exact path. After giving the exact symmetries of the geodesic equations, two different approaches to study the approximate symmet…
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48 results for “geodesic symmetries”
The paper analyzes symmetries of Vaidya-Bonner geodesics.
problem Investigating invariance properties of Vaidya-Bonner geodesics.
method Classification of Lie point symmetries and Noether symmetries, determination of optimal system of subalgebras.
result Determination of optimal system of subalgebras for Vaidya-Bonner 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 on symmetries in geometric structures, finding unique symmetries imply affine symmetry.
problem Investigating symmetries in geometric structures.
method Analyzing (local) automorphisms of parabolic geometries.
result Many parabolic geometries have at most one generalized geodesic symmetry at points with non-zero harmonic curvature.
The paper characterizes strong Hamel functions using symmetries and proves their preservation properties.
problem Characterizing strong Hamel functions and their symmetries in Finsler spaces.
method Analyzing geodesic spray, strong dual symmetries, and strong dynamical symmetries.
result Strong Hamel functions can be characterized in terms of strong dual symmetries and strong dynamical symmetries.
Study complex structures on geodesic and harmonic map spaces.
problem Existence and uniqueness of compatible complex structures.
method Definition and study of adapted complex and involutive structures.
result Existence and uniqueness of such structures are established.
Study geodesic diameter on surfaces with special symmetry.
problem Geodesic diameter on surfaces with involutive isometry.
method Analyzes surfaces with specific symmetry properties.
result Determines geodesic diameter for these surfaces.
Researchers compute c-projective symmetry algebras for Kähler surfaces.
problem Understanding symmetries in Kähler surfaces.
method Defined and analyzed c-projective vector fields and computed their symmetries.
result Computed c-projective symmetry algebras for Kähler surfaces with essential c-projective vector fields.
Functions with constant geodesic X-ray transform are restricted to manifolds with specific geometrical properties.
problem Existence of functions with constant geodesic X-ray transform on manifolds.
method Analyzing the geometrical properties of manifolds based on the existence of such functions.
result Functions with constant geodesic X-ray transform impose specific geometrical restrictions on the manifold.
Study of control problems on Carnot groups with SO(3) symmetry using geometric algebra.
problem Control problems on Carnot groups with SO(3) symmetry.
method Geometric algebra approach to understand geodesics and develop a control algorithm.
result New algorithm for local control developed.
Study of 2D metrics with one projective symmetry leading to superintegrable systems.
problem Classifying 2D metrics with one projective symmetry and their integrable properties.
method Analyzing projective connections, partial differential equations, and geodesic flows.
result Superintegrable systems are parametrized by the 2-sphere, except for 6 exceptional points.
Geodesic rays constructed in Kähler metrics with torus symmetry.
problem Constructing geodesic paths in Kähler metrics with symmetry.
method Mixed polarization, complex structures, geodesic rays.
result Geodesic rays in the space of Kähler metrics.
Proves existence of solutions with concentrated energy in 2+1 spacetime.
problem Existence of solutions with concentrated energy in 2+1 spacetime.
method Direct treatment of 2+1 Einstein equations, novel scaling, Klainerman-Sobolev inequality.
result Uniform finite-time existence of solutions with positive incoming energy.
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.
In this paper, a complete analysis of symmetries and conservation laws for the charged squashed Kaluza--Klein black hole spacetime in a Riemannian space is discussed. First, a comprehensive group analysis of the underlying space-time metric using Lie point symmetries are presented and then it the -dimensional optima…
The paper analyzes Lie symmetries in a specific geometric context.
problem Analyzing Lie symmetries in a canonical connection with a special Lie algebra structure.
method Formulated Lie symmetries for a general linear connection, then applied to the canonical connection with a codimension one abelian nilradical.
result Conditions determining Lie symmetries in the specified geometric context are completely integrated.
We study the Jones and Tod correspondence between selfdual conformal 4-manifolds with a conformal vector field and abelian monopoles on Einstein-Weyl 3-manifolds, and prove that invariant complex structures correspond to shear-free geodesic congruences. Such congruences exist in abundance and so provide a tool for cons…
Study local control in a 7D quaternionic Heisenberg group.
problem Optimizing geodesics in a 7D quaternionic Heisenberg group.
method Matrix representation and analysis of sub-Riemannian structure symmetries.
result Impact of symmetries on geodesic optimality.
Hyperbolic knots decompose into prism orbifolds.
problem Understanding hyperbolic knot complements and their geometric properties.
method Analyzing knot complements as quotients of by discrete groups of reflections in polyhedra with triangular prism combinatorial type.
result Knot complements decompose into hidden symmetries and contain closed, embedded, totally geodesic surfaces.
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.
Study shows no global symmetries for Kähler metrics on compact manifolds.
problem Identifying symmetries in the space of Kähler metrics on compact manifolds.
method Proving isometries are induced by biholomorphisms and anti-biholomorphisms, analyzing Mabuchi's metric and its completion.
result No global symmetries exist for Kähler metrics on compact manifolds, including for Mabuchi's metric.
This study explores magnetic trajectories on the Heisenberg group, finding symmetries and solutions.
problem Understanding magnetic geodesics on the Heisenberg group with invariant Lorentz force.
method Analyzing the Heisenberg Lie group with a non-commutative product, deriving magnetic equations, identifying symmetries, and solving variational problems.
result Magnetic trajectories are solutions to a variational problem, providing explicit examples of Lagrangians.
Symetric Monopolesmath.DG
We discuss Bogomolny monopoles of arbitrary charge invariant under various symmetry groups. The analysis is largely in terms of the spectral curves, the rational maps, and the Nahm equations associated with monopoles. We consider monopoles invariant under inversion in a plane, monopoles with cyclic symmetry…
Sub-Riemannian Geodesics on math.DG
Researchers find shortest paths on a special group structure.
problem Finding shortest paths on a specific group structure.
method Symmetry reduction to a simpler problem, then solving on a quotient space.
result Explicitly described length-minimizing geodesics.
The horizon and geodesic structure of static configurations generated by anisotropic conformal transforms of the Schwarzschild metric is analyzed. We construct the maximal analytic extension of such off--diagonal vacuum metrics and conclude that for small deformations there are different classes of vacuum solutions of …
The paper studies geodesics on an icosahedron's Riemann surface.
problem Geodesics on a icosahedron's Riemann surface.
method Constructs a Riemann surface and analyzes geodesics on it.
result Discovers properties of geodesics under a flat metric.
We establish a bijective correspondence between affine connections and a class of semi-holonomic jets of local diffeomorphisms of the underlying manifold called symmetry jets in the text. The symmetry jet corresponding to a torsion free connection consists in the family of -jets of the geodesic symmetries. Conversel…
We consider robust covariance estimation with group symmetry constraints. Non-Gaussian covariance estimation, e.g., Tyler scatter estimator and Multivariate Generalized Gaussian distribution methods, usually involve non-convex minimization problems. Recently, it was shown that the underlying principle behind their succ…
A 3-parameter family of helical tubular surfaces obtained by screw revolving a circle provides a useful pedagogical example of how to study geodesics on a surface that admits a 1-parameter symmetry group, but is not as simple as a surface of revolution like the torus which it contains as a special case. It serves as a …
Study geodesics on orbifolds with group actions.
problem Existence of closed magnetic geodesics on orbifolds.
method Investigate geodesic flows on orbifolds with invariant metrics and locally-free group actions.
result Existence of closed magnetic geodesics on quotient orbifolds.
We study supersymmetric probe M5-branes in the AdS_4 solution that arises from M5-branes wrapped on a hyperbolic 3-manifold M_3. This amounts to introducing internal defects within the framework of the 3d-3d correspondence. The BPS condition for a probe M5-brane extending along all of AdS_4 requires it to wrap a surfac…
The paper explores the geometric properties of fluid flows and their symmetries.
problem Understanding the geometric properties of fluid flows and their symmetries.
method Analyzing the Euler equation and its relation to geodesic flows on groupoids of multiphase diffeomorphisms.
result Generalized flows, multiphase fluids, and vortex sheets are all geodesics on certain groupoids of multiphase diffeomorphisms.
Study calculates heat coefficients from geodesic polygon corners.
problem Heat coefficients from geodesic polygon corners.
method Formulas for cone points contributions to heat trace.
result Determines Laplacian of Gauss curvature at corners.
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.
We make use of a symmetry reduction technique called Routh reduction to show that the solutions of the Euler-Lagrange equations of a strongly convex autonomous Lagrangian which lie on a specific energy level can be thought of as geodesics of an associated Finsler function.
Researchers develop a new method for analyzing shape changes in medical data.
problem Analyzing continuous transformations like shape changes in medical datasets.
method Geodesic regression using an affine connection setting on Lie groups.
result Efficient fixed point algorithm for computing geodesic relationships.
Sharp inequalities and symmetries on Riemannian surfaces quantified.
problem Understanding symmetries and asymmetries in Riemannian surfaces.
method Introducing scattering energy to measure asymmetry and proving isoperimetric inequalities.
result Sharp quantitative isoperimetric inequalities and domains with vanishing scattering energy characterized.
Study scattering rigidity on stationary manifolds using geodesics.
problem Scattering rigidity on standard stationary manifolds.
method Use Hamiltonian reduction to relate to -systems.
result New rigidity results for stationary manifolds.
Constructs minimal annuli with free boundary in hyperbolic 3-space.
problem Finding minimal surfaces with boundary in hyperbolic geometry.
method Constructs families of non-rotational minimal annuli with shared symmetry.
result Bifurcates from hyperbolic catenoids, forming a countable collection.
Zermelo deformation preserves geodesics and curvature in Finsler metrics.
problem Behavior of geodesics and curvature in Finsler metrics under Zermelo deformation.
method Zermelo deformation with Killing vector fields.
result Zermelo deformation preserves local symmetry in locally symmetric Finsler metrics.
Examining orbits ending in binary collisions for three equal masses under an inverse cube force.
problem Analyzing orbits ending in binary collisions for three equal masses under an inverse cube force.
method Reparametrizing orbits as geodesics on a negatively curved metric on a pair of pants.
result Visibility properties of negatively curved surfaces describe orbits beginning or ending in binary collisions.
In this paper we prove a symmetry result on submanifolds of codimension one in a n + 1-dimensional space form, related to the geodesic distance function and to the normal curvature of some fixed vector field. As applications we will prove sphere characterization type theorems for Kahler manifolds endowed with a toric g…
Study of Ricci flow convergence on surfaces with boundary.
problem Convergence of singular solutions to Ricci flow on compact surfaces with boundary.
method Subsequential convergence analysis of Ricci flow with prescribed geodesic curvature.
result Convergence does not depend on the sign of geodesic curvature of the boundary in the case of rotational symmetry.
Flat fully augmented links with homeomorphic complements are equivalent.
problem Determining equivalence of flat fully augmented links based on their complements.
method Careful analysis of totally geodesic surfaces and cusps in link complements and their behavior under homeomorphism.
result Complete classification of flat fully augmented link complements with multiple reflection surfaces and symmetries.
The paper is a study of geodesic in two-dimensional pseudo-Riemannian metrics. Firstly, the local properties of geodesics in a neighborhood of generic parabolic points are investigated. The equation of the geodesic flow has singularities at such points that leads to a curious phenomenon: geodesics cannot pass through s…
Formula proves symmetry breaking operators for differential forms.
problem Symmetry breaking operators between differential forms on spheres and their hyperplanes.
method Explicit residue formula for meromorphic continuation of operators.
result Simple construction of symmetry breaking operators and determination of zeros.
We develop a new approach, based on quantization methods, to study higher symmetries of invariant differential operators. We focus here on conformally invariant powers of the Laplacian over a conformally flat manifold and recover results of Eastwood, Leistner, Gover and Šilhan. In particular, conformally equivariant qu…
Weaved helices form mechanically stable 3D structures.
problem Creating stable 3D structures from helical elements.
method Exploiting screw symmetry and invariant cylindrical rod packing to form triply periodic arrangements.
result Demonstrated nineteen triply periodic arrangements of interwoven helices.