  We study the persistence of quadratic estimates related to the Kato square
root problem across a change of metric on smooth manifolds by defining a class
of Riemannian-like metrics that are permitted to be of low regularity and
degenerate on sets of measure zero. We also demonstrate how to transmit
quadratic estimates between manifolds which are homeomorphic and locally
bi-Lipschitz. As a consequence, we demonstrate the invariance of the Kato
square root problem under Lipschitz transformations of the space and obtain
solutions to this problem on functions and forms on compact manifolds with a
continuous metric. Furthermore, we show that a lower bound on the injectivity
radius is not a necessary condition to solve the Kato square root problem.
