We show that on every manifold, every conformal class of semi-Riemannian metrics contains a
metric g such that each kth-order covariant derivative of the Riemann tensor of g has bounded absolute
value ak . This result is new also in the Riemannian case, where one can arrange in addition that g
is complete with injectivity and convexity radius ¸ 1. One can even make the radii rapidly increasing
and the functions ak 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.
Moreover, we explain a general principle that can be used to obtain analogous results for Riemannian
manifolds equipped with arbitrary other additional geometric structures instead of foliations.