  In this paper we obtain local rigidity results for linear Anosov
diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that
given an irreducible linear hyperbolic automorphism $L$ with simple real
eigenvalues with distinct absolute values, any small perturbation preserving
the volume and with the same Lyapunov exponents is smoothly conjugate to $L$.
  We also obtain rigidity results for skew products over Anosov
diffeomorphisms. Given a volume preserving partially hyperbolic skew product
diffeomorphism $f_0$ over an Anosov automorphism of the 2-torus, we show that
for any volume preserving perturbation $f$ of $f_0$ with the same average
stable and unstable Lyapunov exponents, the center foliation is smooth.
