  We study the Kasparov product on (possibly non-compact and incomplete)
Riemannian manifolds. Specifically, we show on a submersion of Riemannian
manifolds that the tensor sum of a regular vertically elliptic operator on the
total space and an elliptic operator on the base space represents the Kasparov
product of the corresponding classes in KK-theory. This construction works in
general for symmetric operators (i.e. without assuming self-adjointness), and
extends known results for submersions with compact fibres. The assumption of
regularity for the vertically elliptic operator is not always satisfied, but
depends on the topology and geometry of the submersion, and we give explicit
examples of non-regular operators. We apply our main result to obtain a
factorisation in unbounded KK-theory of the fundamental class of a Riemannian
submersion, as a Kasparov product of the shriek map of the submersion and the
fundamental class of the base manifold.
