  An integral geometric curvature is defined as the index expectation K(x) =
E[i(x)] if a probability measure m is given on vector fields on a Riemannian
manifold or on a finite simple graph. Such curvatures are local, satisfy
Gauss-Bonnet and are independent of any embedding in an ambient space. While
realizing constant Gauss-Bonnet-Chern curvature is not possible in general
already for 4-manifolds, we prove that for compact connected manifolds,
constant curvature K_m can always be realized with m supported on Morse
gradient fields. We give examples of finite simple graphs which do not allow
for any constant m-curvature and prove that for one-dimensional connected
graphs, there is a convex set of constant curvature configurations with
dimension of the first Betti number of the graph. In particular, there is
always a unique constant curvature solution for trees.
