The geometry on the slope of a mountain

Abstract

The geometry on a slope of a mountain is the geometry of a Finsler metric, called here the slope metric. We study the existence of globally defined slope metrics on surfaces of revolution as well as the geodesic's behavior. A comparison between Finslerian and Riemannian areas of a bounded region is also studied.

Finslerian indicatrices as algebraic curves and surfaces

“…that is a Minkowski slope metric (see [6], this approach is sometimes called the Okubo's method). By smoothly moving this Minkowski norm on a 2-dimensional smooth manifold M we get the usual slope metric on M F = α 2 α−β , where α is a Riemanninan metric M and β a linear 1-form (see our recent paper [3] for a study of the slope metric on a surface of revolution).…”

“…For instance, recall that the Randers and Kropina metrics are obtained by a rigid translation of the unit sphere such that the origin is enclosed by it or it is included in its boundary, respectively. We point out that Kropina metrics are actually conic Finsler metrics (see [11], [3] for details).…”

“…Another similar example of Finsler metric is the slope metric (see [3], [6]), where, in the 2 dimensional case, the curve indicatrix is a limaçon. The associated Finsler norm is written in the general form F = α 2 α−β and called the slope metric (or a Matsumoto metric), where α = a ij y i y j is a Riemannian metric and β = b i y j is a linear 1-form.…”

“…The associated Finsler norm is written in the general form F = α 2 α−β and called the slope metric (or a Matsumoto metric), where α = a ij y i y j is a Riemannian metric and β = b i y j is a linear 1-form. In [3] we have studied the geometry of the slope metric globally induced on a surface of revolution.…”