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We prove global smooth classification results for TNS totally Anosov Z^kactions on general compact manifolds, under each one of the followingconditions: joint integrability, resonance-free or Lyapunov pinching condition.Unlike the previous results, we do not require any uniform quasiconformality orpinching condition of action elements on coarse Lyapunov distributions, nor dowe have any restriction on the dimension of coarse Lyapunov distributions. Themain novelty is in proving a new standard form of the derivative cocycle forany TNS totally Anosov Z^k action on general manifold. A main idea is to createa new mechanism called a non-uniform redefining argument to prove continuity ofgeneral dynamically-defined object, which should apply to more general rigidityproblems in dynamical systems.

On classification of higher rank Anosov actions on compact manifold