arXiv research

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…

The Virasoro-Bott group endowed with the right-invariant $L^2$-metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing geodesic distance.

The paper connects Schrödinger equations to geodesics on a 2-surface.

Study magnetic geodesics on Kähler potentials using variational methods.

Given a space it is easy to obtain the system of geodesic equations on it. In this paper the inverse problem of reconstructing the space from the geodesic equations is addressed. A procedure is developed for obtaining the metric tensor from the Christoffel symbols. The procedure is extended for determining if a second …

In 3D, conformal geodesics are variational.

Geodesic interpretation of global quasi-geostrophic equations on sphere.

Solves geodesic equations on specific metrics.

Proves regularity of geodesic equation on Hermitian manifolds.

Solves geodesic equations on special Kähler manifolds, proving global regularity.

We bring together those systems of hydrodynamical type that can be written as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups with right invariant $L^2$ or $H^1$ metrics. We present their formal derivation starting from Euler's equation, the first order equation satisfied by the ri…

Solves geodesic equations on specific metrics types.

We study the evolution equations for a regularized version of Dirac-geodesics, which are the one-dimensional version of Dirac-harmonic maps. We show that for the regularization being sufficiently large, the evolution equations subconverge to a regularized Dirac-geodesic. In the end, we discuss the limiting process of r…

Classifies geodesic vectors in low-dimensional Lie algebras.

We study weak geodesics in the space of potentials for the deformed Hermitian-Yang-Mills equation. The geodesic equation can be formulated as a degenerate elliptic equation, allowing us to employ nonlinear Dirichlet duality theory, as developed by Harvey-Lawson. By exploiting the convexity of the level sets of the Lagr…

Let $M$ be a smooth manifold and $\mathcal{S}$ a semi-spray defined on a sub-bundle $\mathcal{C}$ of the tangent bundle $TM$. In this work it is proved that the only non-trivial $k$-jet approximation to the exact geodesic deviation equation of $\mathcal{S}$, linear on the deviation functions and invariant under an spec…

This paper is devoted to the regularity analysis of a geodesic equation in the space of Sasakian metrics. Firstly, we reduce the geodesic equation in the space of Sasakian metrics to a Dirichlet problem of degenerate complex Monge-Ampére type eqution on the Kähler cone; secondly, we obtain a priori etimates for the abo…

We propose a new two-component geodesic equation with the unusual property that the underlying space has constant positive curvature. In the special case of one space dimension, the equation reduces to the two-component Hunter-Saxton equation.

We study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the $SO(3)$--invariant gravitational instantons. On a hyper--Kähler four--manifold the conformal geodesic equations reduce to geodesic equations of a charged particle moving in a constant self--dual magnetic field. In…

Variationality of conformal geodesics fails in higher dimensions.

Geometrically interprets two equations, showing their equivalence and providing solutions.

In this article we investigate a first order reparametrization-invariant Sobolev metric on the space of immersed curves. Motivated by applications in shape analysis where discretizations of this infinite-dimensional space are needed, we extend this metric to the space of Lipschitz curves, establish the wellposedness of…

Many important equations of mathematical physics arise geometrically as geodesic equations on Lie groups. In this paper, we study an example of a geodesic equation, the two-component Hunter-Saxton (2HS) system, that displays a number of unique geometric features. We show that 2HS describes the geodesic flow on a manifo…

Developed a new formalism to describe Riemannian geometries using geodesic flow bundles.

Geodesics on extended Siegel-Jacobi upper half-plane determined.

Study of complex Hessian equations using subharmonic functions and geodesics.

Connections between Lie derivatives and the deviation equation has been investigated in spaces with affine connection. The deviation equations of the geodesics as well as deviation equations of non-geodesics trajectories have been obtained on this base. This is done via imposing certain conditions on the Lie derivative…

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 $(\bar{M}, \bar{g})$. The natural splitting of the tangent bundle of $…

Study confirms geodesic connectivity and rooftop envelopes in complex Monge-Ampère equation domains.

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…

It is shown that the geodesic rays constructed as limits of Bergman geodesics from a test configuration are always of class $C^{1,α}, 0<α<1$. An essential step is to establish that the rays can be extended as solutions of a Dirichlet problem for a Monge-Ampere equation on a Kaehler manifold which is compact.

The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion. In this work, we extend these investigations to characterize curves that lie on a…

We derive the 2-component Camassa-Holm equation and corresponding N=1 super generalization as geodesic flows with respect to the $H^1$ metric on the extended Bott-Virasoro and superconformal groups, respectively.

Smooth solutions found for hydrodynamic equations.

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…

The paper introduces new equations in Kähler geometry and proves their solutions and convexity.

In this note we find a 6-dimensional h-spaces of the $[(21...1)(21...1)...(1...1)]$ type and then determine quadratic first integrals of the geodesic equations of these h-spaces.

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…

New equations reveal how cylinder power in progressive lenses depends on geodesic curvature.

We study Sobolev-type metrics of fractional order $s\geq0$ on the group $\Diff_c(M)$ of compactly supported diffeomorphisms of a manifold $M$. We show that for the important special case $M=S^1$ the geodesic distance on $\Diff_c(S^1)$ vanishes if and only if $s\leq\frac12$. For other manifolds we obtain a partial chara…

New variational principles found for conformal geodesics.

S. Donaldson introduced a metric on the space of volume forms, with fixed total volume on any compact Riemmanian manifold. With this metric, the space of volume forms formally has non-positive curvature. The geodesic equation is a fully nonlinear degenerate elliptic equation. We solve the geodesic equation and its pert…

Study geodesic equation on mixed-volume forms on balanced manifolds, proving existence of solutions.

Study the geometry of hydrodynamics equations using diffeomorphism groups.

We show that the degenerate special Lagrangian equation, recently introduced by Rubinstein-Solomon, induces a global equation on every Riemannian manifold, and that for certain associated geometries this equation governs, as it does in the Euclidean setting, geodesics in the space of positive Lagrangians. For example, …

This work proposes a model for geodesic distances and flows on manifolds.

The geometric approach to optimal transport and information theory has triggered the interpretation of probability densities as an infinite-dimensional Riemannian manifold. The most studied Riemannian structures are Otto's metric, yielding the $L^2$-Wasserstein distance of optimal mass transport, and the Fisher--Rao me…

Study rigidity of geodesic balls on manifolds with boundary.