Related papers: On General Plane Fronted Waves. Geodesics
Invariant geodesic orbit Finsler $(\alpha,\beta)$ metrics $F$ which arise from Riemannian geodesic orbit metrics $\alpha$ on spheres are determined. The relation of Riemannian geodesic graphs with Finslerian geodesic graphs proved in a…
An important aspect of General Relativity is to study properties of geodesics. A useful tool for describing geodesic behavior is the geodesic deviation equation. It allows to describe the tidal properties of gravitating objects through the…
We investigate typical behavior of geodesics on a closed flat surface $S$ of genus $g\geq 2$. We compare the length quotient of long arcs in the same homotopy class with fixed endpoints for the flat and the hyperbolic metric in the same…
We show that a simply connected Riemannian homogeneous space M which admits a totally geodesic hypersurface F is isometric to either (a) the Riemannian product of a space of constant curvature and a homogeneous space, or (b) the warped…
The aim of this paper is to review and complete the study of geodesics on G\"odel type spacetimes initiated in [8] and improved in [2] of the References. In particular, we prove some new results on geodesic connectedness and geodesic…
We study geodesics in the complete family of expanding impulsive gravitational waves propagating in spaces of constant curvature, that is Minkowski, de Sitter and anti-de Sitter universes. Employing the continuous form of the metric we…
We give detailed exposition of modern differential geometry from global coordinate independent point of view as well as local coordinate description suited for actual computations. In introduction, we consider Euclidean spaces and different…
We propose a new strong Riemannian metric on the manifold of (parametrized) embedded curves of regularity $H^s$, $s\in(3/2,2)$. We highlight its close relationship to the (generalized) tangent-point energies and employ it to show that this…
We study geodesics in the Brownian map $(\mathcal{S},d,\nu)$, the random metric measure space which arises as the Gromov-Hausdorff scaling limit of uniformly random planar maps. Our results apply to all geodesics including those between…
In the present paper we study the global behaviour of geodesics on a Randers metric, defined on a topological cylinder, obtained as the solution of the Zermelo's navigation problem. Our wind is not necessarily a Killing field. In special we…
We find necessary and sufficient conditions for the foliation defined by level sets of a function f(x_{1},...,x_{n}) to be totally geodesic in a torsion-free connection and apply them to find the conditions for d-webs of hypersurfaces to be…
The geometrical diffraction theory, in the sense of Keller,is here reconsidered as an obstacle problem in the Riemannian geometry. The first result is the proof of the existence and the analysis of the main properties of the diffracted…
It is well known in general relativity that trajectories of Hamiltonian systems lift to geodesics of pp-wave spacetimes, an example of a more general phenomenon known as the "Eisenhart lift." We review and expand upon the benefits of this…
This is the second part of a two--part series investigating bifurcation phenomena in autonomous Lagrangian systems and geodesic flows on Finsler and Riemannian manifolds. Building upon the abstract bifurcation theorems established in…
Motivated by the analogies between the projective and the almost quaternionic geometries, we study the generalized planar curves and mappings. We follow, recover, and extend the classical approach as developed by Mikes and Sinyukov. Then we…
Generalized Lagrangian mean theories are used to analyze the interactions between mean flows and fluctuations, where the decomposition is based on a Lagrangian description of the flow. A systematic geometric framework was recently developed…
We construct convex bodies that can be "captured by nets." More precisely, for each dimension $n \geq 2$, we construct a family of Riemannian $n$-spheres, each with a stable geodesic net, which is a stable 1-dimensional integral varifold.…
The geodesic as well as the geodesic deviation equation for impulsive gravitational waves involve highly singular products of distributions $(\theta\de$, $\theta^2\de$, $\de^2$). A solution concept for these equations based on embedding the…
We study geodesic motion in expanding spherical impulsive gravitational waves propagating in a Minkowski background. Employing the continuous form of the metric we find and examine a large family of geometrically preferred geodesics. For…
We study the geometry of the cuspidal edge $M$ in $\mathbb R^3$ derived from its contact with planes and lines (referred to as flat geometry). The contact of $M$ with planes is measured by the singularities of the height functions on $M$.…