Boundary rigidity for Randers metrics

If a non-reversible Finsler norm is the sum of a reversible Finsler norm and a closed 1-form, then one can uniquely recover the 1-form up to potential fields from the boundary distance data. We also show a boundary rigidity result for Randers metrics where the reversible Finsler norm is induced by a Riemannian metric which is boundary rigid. Our theorems generalize Riemannian boundary rigidity results to some non-reversible Finsler manifolds. We provide an application to seismology where the seismic wave propagates in a moving medium.

On geodesically reversible Finsler manifolds

A Finsler manifold is said to be geodesically reversible if the reversed curve of any geodesic remains a geometrical geodesic. Well-known examples of geodesically reversible Finsler metrics are Randers metrics with closed [Formula: see text]-forms. Another family of well-known examples are projectively flat Finsler metrics on the [Formula: see text]-sphere that have constant positive curvature. In this paper, we prove some geometrical and dynamical characterizations of geodesically reversible Finsler metrics, and we prove several rigidity results about a family of the so-called Randers-type Finsler metrics. One of our results is as follows: let [Formula: see text] be a Riemannian–Finsler metric on a closed surface [Formula: see text], and [Formula: see text] be a small antisymmetric potential on [Formula: see text] that is a natural generalization of [Formula: see text]-form (see Sec. 1). If the Randers-type Finsler metric [Formula: see text] is geodesically reversible, and the geodesic flow of [Formula: see text] is topologically transitive, then we prove that [Formula: see text] must be a closed [Formula: see text]-form. We also prove that this rigidity result is not true for the family of projectively flat Finsler metrics on the [Formula: see text]-sphere of constant positive curvature.

Geometry of left-invariant Randers metric on the Heisenberg groups

PurposeIn this paper, we consider the Heisenberg groups which play a crucial role in both geometry and theoretical physics.Design/methodology/approachIn the first part, we retrieve the geometry of left-invariant Randers metrics on the Heisenberg group H2n+1, of dimension 2n + 1. Considering a left-invariant Randers metric, we give the Levi-Civita connection, curvature tensor, Ricci tensor and scalar curvature and show the Heisenberg groups H2n+1 have constant negative scalar curvature.FindingsIn the second part, we present our main results. We show that the Heisenberg group H2n+1 cannot admit Randers metric of Berwald and Ricci-quadratic Douglas types. Finally, the flag curvature of Z-Randers metrics in some special directions is obtained which shows that there exist flags of strictly negative and strictly positive curvatures.Originality/valueIn this work, we present complete Reimannian geometry of left invarint-metrics on Heisenberg groups. Also, some geometric properties of left-invarainat Randers metrics will be studied.

Weighted projective Ricci curvature in Finsler geometry

In this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted projective Ricci flat curvature. Finally, we show that every projectively flat metric with isotropic weighted projective Ricci and isotropic S-curvature is a Kropina metric or Randers metric.

A class of Randers metrics of scalar flag curvature

One of the most important problems in Finsler geometry is to classify Finsler metrics of scalar flag curvature. In this paper, we study the classification problem of Randers metrics of scalar flag curvature. Under the condition that [Formula: see text] is a Killing 1-form, we obtain some important necessary conditions for Randers metrics to be of scalar flag curvature.

Geodesic orbit Randers metrics on spheres

Studying geodesic orbit Randers metrics on spheres, we obtain a complete classification of such metrics. Our method relies upon the classification of geodesic orbit Riemannian metrics on the spheres Sn in [17] and the navigation data in Finsler geometry. We also construct some explicit U(n + 1)-invariant metrics on S2n+1 and Sp(n + 1)U(1)-invariant metrics on S4n+3.

A Zermelo navigation problem with a vortex singularity

Helhmoltz-Kirchhoff equations of motions of vortices of an incompressible fluid in the plane define a dynamics with singularities and this leads to a Zermelo navigation problem describing the ship travel in such a field where the control is the heading angle. Considering one vortex, we define a time minimization problem, geometric frame being the extension of Randers metrics in the punctured plane, with rotational symmetry. Candidates as minimizers are parameterized thanks to the Pontryagin Maximum Principle as extremal solutions of a Hamiltonian vector field. We analyze the time minimal solution to transfer the ship between two points where during the transfer the ship can be either in a strong current region in the vicinity of the vortex or in a weak current region. Analysis is based on a micro-local classification of the extremals using mainly the integrability properties of the dynamics due to the rotational symmetry. The discussion is complex and related to the existence of an isolated extremal (Reeb) circle due to the vortex singularity. Explicit computation of cut points where the extremal curves cease to be optimal is given and the spheres are described in the case where at the initial point the current is weak.