  We establish the factorization of Dirac operators on Riemannian submersions
of compact spin$^c$ manifolds in unbounded KK-theory. More precisely, we show
that the Dirac operator on the total space of such a submersion is unitarily
equivalent to the tensor sum of a family of Dirac operators with the Dirac
operator on the base space, up to an explicit bounded curvature term. Thus, the
latter is an obstruction to having a factorization in unbounded KK-theory. We
show that our tensor sum represents the bounded KK-product of the corresponding
KK-cycles and connect to the early work of Connes and Skandalis.
