  We consider a proper flat fibration with real base and complex fibers. First
we construct odd characteristic classes for such fibrations by a method that
generalizes constructions of Bismut-Lott. Then we consider the direct image of
a fiberwise holomorphic vector bundle, which is a flat vector bundle on the
base. We give a Riemann-Roch-Grothendieck theorem calculating the odd real
characteristic classes of this flat vector bundle.
