  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.
