Since their introduction by Thurston, measured geodesic laminations on hyperbolic surfaces occur in many contexts. In this survey, we give a generalization of geodesic laminations on surfaces endowed with a half-translation structure (that is a singular flat surface with holonomy $\{\pm \mathrm{Id}\}$), called flat laminations, and we define transverse measures on flat laminations similar to transverse measures on hyperbolic laminations, taking into account that the images of the leaves of a flat lamination are in...

Consider a sub-riemannian geometry(U,D,g) where U is a neighborhood of 0 in R3, D is a Martinet type distribution identified to ker ω, ω being the 1-form: $\omega =dz-\frac{y^2}{2}dx$, q=(x,y,z) and g is a metric on D which can be taken in the normal form: $g=a\left(q\right)d{x}^2+c\left(q\right)d{y}^2$, a=1+yF(q), c=1+G(q), $G_{{|}_{x=y=0}}=0$. In a previous article we analyze the flat case: a=c=1; we describe the conjugate and cut loci, the sphere and the wave front. The objectif of this article is to provide a geometric and computational framework to analyze the general case....

For a precompact subset K of a metric space and ε > 0, the covering number N(K,ε) is defined as the smallest number of balls of radius ε whose union covers K. Knowledge of the metric entropy, i.e., the asymptotic behaviour of covering numbers for (families of) metric spaces is important in many areas of mathematics (geometry, functional analysis, probability, coding theory, to name a few). In this paper we give asymptotically correct estimates for covering numbers for a large class of homogeneous...

Given a unital C*-algebra $$𝒜$$ and a right C*-module $$𝒳$$ over $$𝒜$$ , we consider the problem of finding short smooth curves in the sphere $${𝒮}_{𝒳}$$ = x ∈ $$𝒳$$ : 〈x, x〉 = 1. Curves in $${𝒮}_{𝒳}$$ are measured considering the Finsler metric which consists of the norm of $$𝒳$$ at each tangent space of $${𝒮}_{𝒳}$$ . The initial value problem is solved, for the case when $$𝒜$$ is a von Neumann algebra and $$𝒳$$ is selfdual: for any element x 0 ∈ $${𝒮}_{𝒳}$$ and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ $${ℒ}_{𝒜}\left(𝒳\right)$$ , Z* = −Z and ∥Z∥ ≤ π, such...

A geodesic of a homogeneous Riemannian manifold $(M=G/K,g)$ is called homogeneous if it is an orbit of an one-parameter subgroup of $G$. In the case when $M=G/H$ is a naturally reductive space, that is the $G$-invariant metric $g$ is defined by some non degenerate biinvariant symmetric bilinear form $B$, all geodesics of $M$ are homogeneous. We consider the case when $M=G/K$ is a flag manifold, i.eȧn adjoint orbit of a compact semisimple Lie group $G$, and we give a simple necessary condition that $M$ admits a non-naturally reductive...

Existence and uniqueness theorems for weak solutions of a complex Monge-Ampère equation are established, extending the Bedford-Taylor pluripotential theory. As a consequence, using the Tian-Yau-Zelditch theorem, it is shown that geodesics in the space of Kähler potentials can be approximated by geodesics in the spaces of Bergman metrics. Motivation from Donaldson’s program on constant scalar curvature metrics and Yau’s strategy of approximating Kähler metrics by Bergman metrics is also discussed....

Given a one-parameter family $\{g_{\lambda }:\lambda \in [a,b]\}$ of semi Riemannian metrics on an n-dimensional manifold M, a family of time-dependent potentials $\{V_{\lambda }:\lambda \in [a,b]\}$ and a family $\{{\sigma }_{\lambda }:\lambda \in [a,b]\}$ of trajectories connecting two points of the mechanical system defined by $(g_{\lambda },V_{\lambda })$, we show that there are trajectories bifurcating from the trivial branch ${\sigma }_{\lambda }$ if the generalized Morse indices $\mu \left({\sigma }_a\right)$ and $\mu \left({\sigma }_b\right)$ are different. If the data are analytic we obtain estimates for the number of bifurcation points on the branch and, in particular, for the number of strictly conjugate...

A survey of recent progress on the multiplicity and stability problems for closed geodesics on Finsler 2-spheres is given.