  We study the differential geometric consequences of our previous result on
the existence of fat triangulations, in conjunction with a result of Cheeger,
M\"{u}ller and Schrader, regarding the convergence of Lipschitz-Killing
curvatures of piecewise-flat approximations of smooth Riemannian manifolds. A
further application to the existence of quasiconformal mappings between
manifolds, as well as an extension of the triangulation result to the case of
almost Riemannian manifolds, are also given. In addition, the notion of fatness
of triangulations and its relation to metric curvature and to excess is
explored. Moreover, applications of the main results, and in particular a
purely metric approach to Regge calculus, are also investigated.
