  In our previous paper \cite{Li2010}, we proved a martingale transform
representation formula for the Riesz transforms on forms over complete
Riemannian manifolds, and proved some explicit $L^p$-norm estimates for the
Riesz transforms on complete Riemannian manifolds with suitable curvature
conditions. In this paper we correct a gap contained in \cite{Li2010} and prove
that the main result obtained in \cite{Li2010} on the $L^p$-norm estimates for
the Riesz transforms on forms remain valid. Moreover, we prove a time reversal
martingale transform representation formula for the Riesz transforms on forms.
Finally, we extend our approach and result to the Riesz transforms acting on
Euclidean vector bundles over complete Riemannian manifolds with suitable
curvature conditions.
