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We consider smooth, complete Riemannian manifolds which are exponentiallylocally doubling. Under a uniform Ricci curvature bound and a uniform lowerbound on injectivity radius, we prove a Kato square root estimate for certaincoercive operators over the bundle of finite rank tensors. These results areobtained as a special case of similar estimates on smooth vector bundlessatisfying a criterion which we call generalised bounded geometry. We provethis by establishing quadratic estimates for perturbations of Dirac typeoperators on such bundles under an appropriate set of assumptions.

The Kato square root problem on vector bundles with generalised bounded geometry