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

Finsler connection preserving the two-vector angle under the indicatrix-inhomogeneous treatment