Fat Triangulations, Curvature and Quasiconformal Mappings

We investigate the interplay between the existence of fat triangulations, PL approximations of Lipschitz-Killing curvatures and the existence of quasiconformal mappings. In particular we prove that if there exists a quasiconformal mapping between two PL or smooth n-manifolds, then their Lipschitz-Killing curvatures are bilipschitz equivalent. An extension to the case of almost Riemannian manifolds, of a previous existence result of quasimeromorphic mappings on manifolds due to the first author is also given.