Finsleroid space with angle and scalar product

Abstract

A systematic approach has been developed to encompass the Minkowski-type extension of Euclidean geometry such that a one-vector anisotropy is permitted, retaining simultaneously the concept of angle. For the respective geometry, the Euclidean unit ball is to be replaced by the body which is convex and rotund and is found on assuming that its surface (the indicatrix extending the unit sphere) is a space of constant positive curvature. We have called the body the Finsleroid in view of its intrinsic relationship with the metric function of Finsler type. The main point of the present paper is the angle coming from geodesics through the cosine theorem, the underlying idea being to derive the angular measure from the solutions to the geodesic equation which prove to be obtainable in simple explicit forms. The substantive items concern geodesics, angle, scalar product, perpendicularity, and two-vector metric tensor. The Finsleroid-space-associated one-vector Finslerian metric function admits in quite a natural way an attractive two-vector extension.