Higher order distance-like functions and Sobolev spaces

We consider complete Riemannian manifolds with a controlled growth of the covariant derivatives of Ricci curvatures up to order $k-2$ and a controlled decay of the injectivity radii. On such manifolds we construct distance-like functions with a control on covariant derivatives up to order $k$. Alternatively, the assumption on the injectivity radii can be replaced with the request of a controlled growth of the full curvature tensor at order $0$. The control in the assumptions occur via non-necessarily polynomial growth functions. This construction largely extend previously known results in various directions, permitting to obtain consequences which are (in a sense) sharp. A first main application is to the study of the density property for Sobolev spaces on Riemannian manifolds, namely the problem of guaranteeing the density of smooth compactly supported function in the Sobolev space $W^{k,p}$. Contrary to all previously known results this can be obtained also on manifolds with possibly unbounded geometry. In the particular case $p=2$, making use of the Weitzenb\"ock formula for a Lichnerowicz Laplacian acting on the space of smooth section of the bundle of $k$-covariant symmetric tensors, we can weaken the assumptions needed to obtain the density property. Namely we prove that the control on the highest order derivative of curvature is not needed in this situation. Beyond the density property we finally highlight some new applications of our results to disturbed Sobolev inequalities, disturbed $L^{p}$-Calder\'on-Zygmund inequalities and the full Omori-Yau maximum principle for the Hessian.