  We prove an extension of Milnor-Wood inequalities to a geometric situation.
We study representations of the fundamental group of a compact manifold into
the isometry group of a product of rank one spaces of the same dimension and
show an upper bound on the volume of the representation. When the target group
is the isometry group of the real hyperbolic space, we show the constance of
the volume under deformations using the Schl\"afli formula and deduce a new and
simple proof of a result of T. Soma; the result is that there are only finitely
many closed hyperbolic three-manifolds dominated by a given closed
three-manifold.
