  The purpose of this paper is to extend our previous work on the variational
formula for the Bismut-Cheeger eta form without the kernel bundle assumption by
allowing the spin$^c$ Dirac operators to be twisted by isomorphic vector
bundles, and to establish the $\mathbb{Z}_2$-graded additivity of the
Bismut-Cheeger eta form. Using these results, we give alternative proofs of the
fact that the analytic index in differential $K$-theory is a well defined group
homomorphism, and the Riemann-Roch-Grothendieck theorem in
$\mathbb{R}/\mathbb{Z}$ $K$-theory.

K-Theory and Homology;Differential Geometry