Fiberwise Volume Decreasing Diffeomorphisms on Product Manifolds

Given a closed connected Riemannian manifold M and a connected Riemannian manifold N , we study fiberwise, i.e. M × {z}, z ∈ N , volume decreasing diffeomorphisms on the product M × 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 fiberwise volume preserving. As a first corollary, we show that the isometries of M × N split. We also study properly discontinuous actions of a discrete group on M × N . In this case, we generalize the first Bieberbach theorem and prove a special case of an extension of Talelli’s conjecture.