Related papers: The geodesic flow on nilmanifolds

We introduce a new family of thermostat flows on the unit tangent bundle of an oriented Riemannian $2$-manifold. Suitably reparametrised, these flows include the geodesic flow of metrics of negative Gauss curvature and the geodesic flow…

The geodesic flow of a Riemannian metric on a compact manifold $Q$ is said to be toric integrable if it is completely integrable and the first integrals of motion generate a homogeneous torus action on the punctured cotangent bundle…

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…

Let $\Sigma$ be a compact quotient of $T_4$, the Lie group of $4 \times 4$ upper triangular matrices with unity along the diagonal. The Lie algebra $t_4$ of $T_4$ has the standard basis $\{X_{ij}\}$ of matrices with $0$ everywhere but in…

We study the geometry of Lie groups $G$ with a continuous Finsler metric, assuming the existence of a subgroup $K$ such that the metric is right-invariant for the action of $K$. We present a systematic study of the metric and geodesic…

We study geodesics flows on curved quantum Riemannian geometries using a recent formulation in terms of bimodule connections and completely positive maps. We complete this formalism with a canonical $*$ operation on noncommutative vector…

We study the geodesic flow of geometrically finite quotients $\Omega/{\Gamma}$ of Hilbert geometries, in particular its recurrence properties. We prove that, under a geometrical assumption on the cusps, the geodesic flow is uniformly…

Two Riemannian manifolds are said to have $C^k$-conjugate geodesic flows if there exist an $C^k$ diffeomorphism between their unit tangent bundles which intertwines the geodesic flows. We obtain a number of rigidity results for the geodesic…

We prove the upper semicontinuity of the measure theoretic entropy for the geodesic flow on complete Riemannian manifolds without focal points and bounded sectional curvature. We then study the relationship between the escape of mass…

We study the projection of the left-invariant sub-Riemannian structure on the 3D Heisenberg group $G$ to the Heisenberg 3D nil-manifold $M$ -- the compact homogeneous space of $G$ by the discrete Heisenberg group. First we describe…

In this paper, we prove Ruelle's inequality for the geodesic flow in non-compact manifolds with Anosov geodesic flow and some assumptions on the curvature. In the same way, we obtain Pesin's formula for Anosov geodesic flow in non-compact…

In this paper, we study the Ricci-Bourguignon flow on higher dimensional classical Heisenberg nilpotent Lie groups and construct a solution of this flow on Heisenberg and quaternion nilpotent Lie groups. In the end, we investigate the…

Various integrable geodesic flows on Lie groups are shown to arise by taking moments of a geodesic Vlasov equation on the group of canonical transformations. This was already known for both the one- and two-component Camassa-Holm systems.…

The geometric evolution equations provide new ways to address a variety of non-linear problems in Riemannian geometry, and, at the same time, they enjoy numerous physical applications, most notably within the renormalization group analysis…

This paper shows how left and right actions of Lie groups on a manifold may be used to complement one another in a variational reformulation of optimal control problems equivalently as geodesic boundary value problems with symmetry. We…

We consider Lie groups equipped with arbitrary distances. We only assume that the distance is left-invariant and induces the manifold topology. For brevity, we call such object metric Lie groups. Apart from Riemannian Lie groups,…

In this note we prove that the Heisenberg group with a left-invariant pseudo-Riemannian metric admits a completely integrable totally geodesic distribution of codimension 1. This is on the contrary to the Riemannian case, as it was proved…

The motion of a rigid body immersed in an incompressible perfect fluid which occupies a three- dimensional bounded domain have been recently studied under its PDE formulation. In particular classical solutions have been shown to exist…

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…

This note constructs a compact, real-analytic, riemannian 4-manifold ({\Sigma}, g) with the properties that: (1) its geodesic flow is completely integrable with smooth but not real-analytic integrals; (2) {\Sigma} is diffeomorphic to $T^2…