Abstract
We show that on every manifold, every conformal class of semi-Riemannian metrics contains a metric such that each th-order covariant derivative of the Riemann tensor of has bounded absolute value . This result is new also in the Riemannian case, where one can arrange in addition that is complete with injectivity and convexity radius 1. One can even make the radii rapidly increasing and the ...
Abstract
We show that on every manifold, every conformal class of semi-Riemannian metrics contains a metric such that each th-order covariant derivative of the Riemann tensor of has bounded absolute value . This result is new also in the Riemannian case, where one can arrange in addition that is complete with injectivity and convexity radius 1. One can even make the radii rapidly increasing and the functions rapidly decreasing at infinity. We prove generalizations to foliated manifolds, where curvature, second fundamental form and injectivity radius of the leaves can be controlled similarly. Still more generally, we introduce the notion of a "flatzoomer": a quantity that involves arbitrary geometric structures and behaves suitably with respect to modifications by a function, e.g. a conformal factor. The results on bounded geometry follow from a general theorem about flatzoomers, which might be applicable in many other geometric contexts involving noncompact manifolds.
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