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We briefly survey some of the recent results concerning the metric behaviorof the invariant foliations for a partially hyperbolic on a three-dimensionalmanifold and propose a conjecture to characterize atomic behavior forconservative partially hyperbolic homotopic to Anosov (DA) on $\mathbb T^3$. Onthe other hand we prove that if one of the invariant foliations (stable, centeror unstable) of a conservative DA on $\mathbb T^3$ is $C^1$ and transverselyabsolutely continuous with bounded Jacobians the Lyapunov exponent on thisdirection is defined everywhere and constant. If the center foliation is thisfoliaion then the DA diffeomophism is smoothly conjugated to a linear Anosov,in particular Anosov. Another consequence of the main theorem is that it doesnot exist a conservative Ma\~n\'e's example.

Lyapunov exponents and regularity of invariant foliations for partially hyperbolic diffeomophisms on $\mathbb T^3$