  Discrete forms of the scalar, sectional and Ricci curvatures are constructed
on simplicial piecewise flat triangulations of smooth manifolds, depending
directly on the simplicial structure and a choice of dual tessellation. This is
done by integrating over volumes which include appropriate samplings of hinges
for each type of curvature, with the integrals based on the parallel transport
of vectors around hinges. Computations for triangulations of a diverse set of
manifolds show these piecewise flat curvatures to converge to their smooth
values. The Ricci curvature also gives a piecewise flat Ricci flow as a
fractional rate of change of edge-lengths, again converging to the smooth Ricci
flow for the manifolds tested.
