  We show that if a Finsler space is conformally automorphic to a Riemannian
space and the automorphism is positively homogeneous with respect to tangent
vectors, then the indicatrix of the Finsler space is a space of constant
curvature. In this case, the Finslerian two-vector angle can explicitly be
found, which gives rise to simple and explicit representation for the
connection preserving the angle in the indicatrix-homogeneous case. The
connection is metrical and the Finsler space is obtainable from the Riemannian
space by means of the parallel deformation. Since also the transitivity of
covariant derivative holds, in such Finsler spaces the metrical non-linear
angle-preserving connection is the respective export of the metrical linear
Riemannian connection. From the commutators of covariant derivatives the
associated curvature tensor is found. In case of the ${\cal FS}$-space, the
explicit example of the conformally automorphic transformation can be
developed, which entails the explicit connection coefficients and the metric
function of the Finsleroid type.
