  We illustrate connections between differential geometry on finite simple
graphs G=(V,E) and Riemannian manifolds (M,g). The link is that curvature can
be defined integral geometrically as an expectation in a probability space of
Poincare-Hopf indices of coloring or Morse functions. Regge calculus with an
isometric Nash embedding links then the Gauss-Bonnet-Chern integrand of a
Riemannian manifold with the graph curvature. There is also a direct
nonstandard approach: if V is a finite set containing all standard points of M
and E contains pairs which are infinitesimally close in the sense of internal
set theory, one gets a finite simple graph (V,E) which gets a curvature which
as a measure corresponds to the standard curvature. The probabilistic approach
is an umbrella framework which covers discrete spaces, piecewise linear spaces,
manifolds or varieties.
