  In this paper we study the existence of positive Lyapunov exponents for three
different types of skew products, whose fibers are compact Riemannian surfaces
and the action on the fibers are by volume preserving diffeomorphisms. These
three types include skew products with a volume preserving Anosov
diffeomorphism on the basis; or with a subshift of finite type on the basis
preserving a measure with product structure; or locally constant skew products
with Bernoulli shifts on the basis. We prove the $C^1$-density and
$C^r$-openess of the existence of positive Lyapunov exponents on a set of
positive measure in the space of such skew products.
