  We consider smooth, complete Riemannian manifolds which are exponentially
locally doubling. Under a uniform Ricci curvature bound and a uniform lower
bound on injectivity radius, we prove a Kato square root estimate for certain
coercive operators over the bundle of finite rank tensors. These results are
obtained as a special case of similar estimates on smooth vector bundles
satisfying a criterion which we call generalised bounded geometry. We prove
this by establishing quadratic estimates for perturbations of Dirac type
operators on such bundles under an appropriate set of assumptions.
