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 Z₂-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 R/Z K-theory.

The author would like to thank Steve Rosenberg for his comments and suggestions for this paper, and Jonathan Kin-Yue Lee, where the idea of Proposition 4.4 is due to him. The author would also like to thank the referee for the helpful comments.