Constant index expectation curvature for graphs or Riemannian manifolds

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.