Eta-invariants, Torsion forms and Flat vector bundles

We present a new proof, as well as a $\mathbf{C}\mathbf{/}\mathbf{Q}$ extension, of the Riemann-Roch-Grothendieck theorem of Bismut-Lott for flat vector bundles. The main techniques used are the computations of the adiabatic limits of $\eta$-invariants associated to the so-called sub-signature operators. We further show that the Bismut-Lott analytic torsion form can be derived naturally from the transgression of the $\eta$-forms appearing in the adiabatic limit computations.