  We show that on every manifold, every conformal class of semi-Riemannian
metrics contains a metric $g$ such that each $k$-th-order covariant derivative
of the Riemann tensor of $g$ has bounded absolute value $a_k$. This result is
new also in the Riemannian case, where one can arrange in addition that $g$ is
complete with injectivity and convexity radius greater than 1. One can even
make the radii rapidly increasing and the functions $a_k$ 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.
