Mathematics

Let $(M,g)$ be a complete non-compact Riemannian manifold. The distance function $r(x)$ from a fixed reference point in general fails to be everywhere differentiable. We seek for geometric assumptions which garantee the existence of a function $H$ on $M$ which is smooth, distance-like (i.e. $\tfrac{r(x)}{C} < H(x) < C r(x)$ outside a compact set) and whose derivatives are bounded up to a certain order. We will present classical results and some more recent answers to this problem. As we will see, distance-like functions permit to prove the density of smooth compactly supported functions in Sobolev spaces on manifolds, and to generalize to $M$ other tools and properties which are well-known in the Euclidean space.