Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)This paper is concerned with the rigidity problem for volume preserving partially hyperbolic diffeomorphisms on [GRAPHICS] homotopic to a linear Anosov on [GRAPHICS] . We characterize the C1+theta conjugacy of such diffeomorphisms in terms of smooth conditions on the stable and unstable foliations. The smooth conditions on the foliations are to be C-1 foliations and transversely absolutely continuous with bounded Jacobians. In particular these conditions for only one direction (stable, centre or unstable) completely determine the Lyapunov exponents.302 189199Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)FAPESP [2011/21214-3, 2012/06553-9]