Given any rectangular polyhedron $3$ -manifold $\mathcal {P}$ tiled with unit cubes, we find infinitely many explicit directions related to cubic algebraic numbers such that all half-infinite geodesics in these directions are uniformly distributed in $\mathcal {P}$ .

We show that geodesics in $\mathbf {H}$ attached to a maximal split torus or a real quadratic torus in $GL_{2, \mathbf {Q}}$ are the only irreducible algebraic curves in $\mathbf {H}$ whose image in $\mathbf {R}^2$ via the j-invariant is contained in an algebraic curve.

The paper surveys open problems and questions related to geodesics defined by Riemannian, Finsler, semi-Riemannian and magnetic structures on manifolds. It is an extended report on problem sessions held during the International Workshop on Geodesics in August 2010 at the Chern Institute of Mathematics in Tianjin.

The problem of differences between the sea mile and the international nautical mile has been analysed. Algorithms for the calculation of sea miles with applications in different sailing methods on the ellipsoid that can be easily incorporated in modern microprocessor controlled navigational devices are proposed. These algorithms can also be employed on an outfit of large-scale nautical charts with a double scale in sea miles and international nautical miles.

We discuss the existence of finite critical trajectories connecting two zeros in certain families of quadratic differentials. In addition, we reprove some results about the support of the limiting root-counting measures of the generalised Laguerre and Jacobi polynomials with varying parameters.

It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied, and the best upper bounds to date are linear in genus, due to a theorem of Buser and Seppälä. The goal of this note is to give a short proof of a linear upper bound that slightly improves the best known bound.

In Carnot groups of step  ≤ 3, all subriemannian geodesics are proved to be normal. Theproof is based on a reduction argument and the Goh condition for minimality of singularcurves. The Goh condition is deduced from a reformulation and a calculus of the end-pointmapping which boils down to the graded structures of Carnot groups.

In this paper we investigate analytic affine control systems \hbox{$\dot{q}$}q̇ = X + uY, u ∈  [a,b] , whereX,Y is an orthonormal frame for a generalized Martinet sub-Lorentzianstructure of order k of Hamiltonian type. We construct normal forms forsuch systems and, among other things, we study the connection between the presence of thesingular trajectory starting at q0 on the boundary of thereachable set from q0 with the minimal number of analyticfunctions needed for describing the reachable set from q0.

It is well known in Kähler geometry that the infinite-dimensional symmetric space of smooth Kähler metrics in a fixed Kähler class on a polarized Kähler manifold is well approximated by finite-dimensional submanifolds of Bergman metrics of height k. Then it is natural to ask whether geodesics in can be approximated by Bergman geodesics in . For any polarized Kähler manifold, the approximation is in the C0 topology. For some special varieties, one expects better convergence: Song and Zelditch proved the C2 convergence for the torus-invariant metrics over toric varieties. In this article, we show that some C∞ approximation exists as well as a complete asymptotic expansion for principally polarized abelian varieties.

The problem of finding geodesics that avoid certain obstacles in negatively curved manifolds has been studied in different situations. In this note we give a generalization of the unclouding theorem of J. Parkkonen and F. Paulin: there is a constant s0=1.534 such that for any Hadamard manifold M with curvature ≤−1 and for any family of disjoint balls or horoballs {Ca}a∈A and for any point p∈M−⋃ a∈ACa if we shrink these balls uniformly by s0 one can always find a geodesic ray emanating from p that avoids the shrunk balls. It will be shown that in the theorem above one can replace the balls by arbitrary convex sets.