  The Finsler spaces in which the tangent Riemannian spaces are conformally
flat prove to be characterized by the condition that the indicatrix is a space
of constant curvature. In such spaces the Finslerian normalized two-vector
angle can be explicated from the respective two-vector angle of the associated
Riemannian space. Therefore the way is opening to propose explicitly the
connection preserving the angle even at the indicatrix-inhomogeneous level,
that is, when the indicatrix curvature value ${\mathcal C}_{\text{Ind.}} $ is
permitted to be an arbitrary smooth function of the indicatrix position point
$x$. The connection obtained is metrical with the deflection part which is
proportional to the gradient of the function $H(x)$ entering the equality
${\mathcal C}_{\text{Ind.}} \equiv H^2.$ Also the connection is
covariant-constant. When the transitivity of covariant derivative is used, from
the commutators of covariant derivatives the associated curvature tensor is
found. Various useful representations have been developed. The Finsleroid space
has been explicitly outlined.
