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We describe a general method of arithmetic coding of geodesics on the modular surface based on a two parameter family of continued fraction transformations studied previously by the authors. The finite rectangular structure of the…

In this paper we study the geodesic flow on nilmanifolds equipped with a left-invariant metric. We write the underlying definitions and find general formulas for the Poisson involution. As an example we develop the Heisenberg Lie group…

In this survey article we gather classical as well as recent results on minimal geodesics of Riemannian or Finsler metrics, giving special attention to the two-dimensional case. Moreover, we present open problems together with some first…

In this article, we consider the geodesic flow on a compact rank $1$ Riemannian manifold $M$ without focal points, whose universal cover is denoted by $X$. On the ideal boundary $X(\infty)$ of $X$, we show the existence and uniqueness of…

In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus $T^2$ for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the…

Consider the geodesic flow on a real-analytic closed hypersurface $M$ of $\mathbb{R}^n$, equipped with the standard Euclidean metric. The flow is entirely determined by the manifold and the Riemannian metric. Typically, geodesic flows are…

We study totally geodesic codimension 1 smooth foliations on Lorentzian manifold. We are in particular interested by the relations between riemannian flows and geodesic foliations. We prove that, up to a 2-cover, any Seifert bundle admit…

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and…

We construct a class of Riemannian metrics in closed surfaces of genus greater than one, having Anosov geodesic flows, and some regions of positive curvature, such that for each such surface, there exists a smooth curve of conformal…

In this paper we prove that if the geodesic flow of a {compact or non-compact} complete manifold without conjugate points is of the Anosov type, then the average of the integral of the sectional curvature along the geodesic is negative and…

We extend two results from the theory of geodesic flows to the magnetic setting on manifolds of arbitrary dimension. First, we investigate the magnetic ray transform and establish a tensor tomography result. Second, we define and analyze…

The set of directions from a quadratic differential that diverge on average under Teichmuller geodesic flow has Hausdorff dimension exactly equal to one-half.

In the present paper we show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This helps us to describe the…

We employ the curve shortening flow to establish three new results on the dynamics of geodesic flows of closed Riemannian surfaces. The first one is the stability, under $C^0$-small perturbations of the Riemannian metric, of certain flat…

Reider's Theorem on the very ampleness of adjoint linear series on a complex projective algebraic surface is extended in two new directions. First, Reider-type inequalities are shown to imply nefness of linear series of the form dH - E on…

Let $M$ be a pinched negatively curved Riemannian orbifold, whose fundamental group has torsion of order $2$. Generalizing results of Sarnak and Erlandsson-Souto for constant curvature oriented surfaces, and with very different techniques,…

We start by constructing a Hilbert manifold T of orientation preserving diffeomorphisms of the circle (modulo the group of bi-holomorphic self-mappings of the disc). This space, which could be thought of as a completion of the universal…

We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces, prove the nonexistence of…

In this paper we show that dynamical and counting results characteristic of negatively-curved Riemannian geometry, or more generally CAT(-1) or rank-one CAT(0) spaces, also hold for geometrically-finite strictly convex projective structures…

We prove that the geodesic flow for the Weil-Petersson metric on the moduli space of Riemann surfaces is ergodic (in fact Bernoulli) and has finite, positive metric entropy.