  We prove that a representation from the fundamental group of a closed surface
of negative Euler characteristic with values in the isometry group of a
Riemannian manifold of sectional curvature bounded by -1 can be dominated by a
Fuchsian representation. Moreover, we prove that the domination can be made
strict, unless the representation is discrete and faithful in restriction to an
invariant totally geodesic 2-plane of curvature -1. When applied to
representations into PSL(2,R) of non-extremal Euler class, our result is a step
forward in understanding the space of closed anti-de Sitter 3-manifolds.
