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