Classical linear maps associated with quantum maps are investigated. It is demonstrated that with the help of SU(n) coherent states, SU(n) tensors fulfilling nonlinear identities can be constructed. Nonlinear maps which evolve like linear maps if initial conditions lie on a manifold which is explicitly given are defined and the analogy with the Kolmogorov-Arnold-Moser theorem is discussed. The problem of geometric quantization is also investigated and a geometric relation between the analysed quantum and classical maps is found.