Finsleroid gives rise to the angle-preserving connection

Abstract

The Finslerian unit ball is called the {\it Finsleroid} if the covering indicatrix is a space of constant curvature. We prove that Finsler spaces with such indicatrices possess the remarkable property that the tangent spaces are conformally flat with the conformal factor of the power dependence on the Finsler metric function. It is amazing but the fact that in such spaces the notion of the two-vector angle defined by the geodesic arc on the indicatrix can readily be induced from the Riemannian space obtained upon the conformal transformation, which opens up the straightforward way to induce also the connection coefficients and the concomitant curvature tensor. Thus, we are successfully inducing the Levi-Civita connection from the Riemannian space into the Finsleroid space, obtaining the isometric connection. The resultant connection coefficients are not symmetric. However, the metricity condition holds fine, that is, the produced covariant derivative of the Finsleroid metric tensor vanishes identically. The particular case underlined by the axial Finsleroid of the ${\mathbf\cF\cF^{PD}_{g}}$-type is explicitly evaluated in detail. Keywords: Finsler metrics, connection, curvature, conformal properties.