The cut locus of a Randers rotational 2-sphere of revolution

Abstract

In the present paper we study structure of the cut locus of a Randers rotational 2-sphere of revolution (M, F = α + β). We show that in the case when Gaussian curvature of the Randers surface is monotone along a meridian the cut locus of a point q ∈ M is a point on a subarc of the opposite half bending meridian or of the antipodal parallel (Theorem 1.1). More generally, when the Gaussian curvature is not monotone along the meridian, but the cut locus of a point q on the equator is a subarc of the same equator, then the cut locus of any point q ∈ M different from poles is a subarc of the antipodal parallel (Theorem 1.2). Some examples are also given at the last section. * Mathematics Subject Classification (2010) : 53C60, 53C22. † Keywords: Randers metrics, 2-sphere of revolution, cut locus, Gaussian curvature.2 The 2-sphere of revolution 2.1 The Riemannian 2-sphere of revolution A compact Riemannian manifold (M, h) homeomorphic to a 2-sphere is called a 2sphere of revolution if M admits a point p, called pole, such that for any two points q 1 , q 2 on M with d h (p, q 1 ) = d h (p, q 2 ), there exists an h-isometry f on M satisfying f (q 1 ) = q 2 , and f (p) = p, where d h (·, ·) denoted the h-Riemannian distance function on M.Let (r, θ) denote geodesic polar coordinates around a pole p of (M, h). The Riemannian metric can be expressed as h = dr 2 + m 2 (r)dθ 2 on M \ {p, q}, where q denotes the unique

A Zermelo navigation problem with a vortex singularity

“…since along the extremal the Hamiltonian is constant and equal to −p 0 . Putting (23) in (22), one has…”

“…Parallel development were obtained in the frame of space navigation where geometric analysis is combined with numerical methods, see again [11] for a general reference for such contributions. In the frame of Zermelo navigation problem with a small current, called Randers metrics, some results were obtained recently for sphere of revolutions [22].…”

Abnormal geodesics in 2D-Zermelo navigation problems in the case of revolution and the fan shape of the small time balls

“…Note that in (2), the current is along the parallels only, which is sufficient to cover the two founding examples. We refer to [11] for a case study in the differential geometric frame, in the case of a weak current, that is Randers problems in Finsler geometry [2], assuming F 0 g < 1.…”

The Geometry of a Randers Rotational Surface with an Arbitrary Direction Wind

“…We will focus on Finsler metrics of Randers type obtained as solutions of the Zermelo's navigation problem, for the navigation data (M, h), where h is the canonical Riemannian metric on the topological cylinder h = dr 2 + m 2 (r)dθ 2 , and W = A(r) ∂ ∂r + B ∂ ∂θ is a vector field on M. Observe that our wind is more general than a Killing vector field, hence our theory presented here is a generalization of the classical study of geodesics and cut locus for Randers metrics obtained as solutions of the Zermelo's navigation problem with Killing vector fields studied in [10,11] (see also [12] for another attempt to generalize the Zermelo's navigation problem). Nevertheless, by taking the wind W in this way, we obtain a quite general Randers metric on M which is a Finsler metric of revolution and whose geodesics and cut locus can be computed explicitly.…”