  The purpose of this paper is to give a proof of the real part of the
Riemann-Roch-Grothendieck theorem for complex flat vector bundles at the
differential form level in the even dimensional fiber case. The proof is,
roughly speaking, an application of the local family index theorem for a
perturbed twisted spin Dirac operator, a variational formula of the
Bismut-Cheeger eta form without the kernel bundle assumption in the even
dimensional fiber case, and some properties of the Cheeger-Chern-Simons class
of complex flat vector bundle.
