Differential structures of Frölicher spaces on tangent curves

We consider the differential-geometric structures of the Frölicher spaces for a singular manifold, which consists of two tangent curves. Calculations for two types of structures lead either to the $\infty$-flatness of the curves, which at a singular point pass from one branch to another, or to the $\infty$-flatness of functions. In the second case, smooth curves can change the branch of motion, their velocity vector at a singular point is zero.