  We solve the Kato square root problem for divergence form operators on
complete Riemannian manifolds that are embedded in Euclidean space with a
bounded second fundamental form. We do this by proving local quadratic
estimates for perturbations of certain first-order differential operators that
act on the trivial bundle over a complete Riemannian manifold with at most
exponential volume growth and on which a local Poincar\'{e} inequality holds.
This is based on the framework for Dirac type operators that was introduced by
Axelsson, Keith and McIntosh.
