  We consider Anosov diffeomorphisms on $\mathbb{T}^3$ such that the tangent
bundle splits into three subbundles $E^s_f \oplus E^{wu}_f \oplus E^{su}_f.$ We
show that if $f$ is $C^r, r \geq 2,$ volume preserving, then $f$ is $C^1$
conjugated with its linear part $A$ if and only if the center foliation
$\mathcal{F}^{wu}_f$ is absolutely continuous and the equality
$\lambda^{wu}_f(x) = \lambda^{wu}_A,$ between center Lyapunov exponents of $f$
and $A,$ holds for $m$ a.e. $x \in \mathbb{T}^3.$ We also conclude rigidity of
derived from Anosov diffeomorphism, assuming an strong absolute continuity
property (Uniform bounded density property) of strong stable and strong
unstable foliations.
