Fiberwise volume decreasing diffeomorphisms on product manifolds

Given a closed connected Riemannian manifold M and a connected Riemannian manifold N, we study fiberwise volume decreasing diffeomorphisms on the product M x N. Our main theorem shows that in the presence of certain cohomological condition on M and N such diffeomorphisms must map a fiber diffeomorphically onto another fiber and are therefore fiber volume preserving. As a first corollary, we show that the isometries of M x N split. We also study properly discontinuous actions of a discrete group on M x N. In this case, we generalize the first Bieberbach theorem and prove a special case of an extension of Talelli's conjecture.