Integral geometric Hopf conjectures

Abstract

The Hopf sign conjecture states that a compact Riemannian 2d-manifold M of positive curvature has Euler characteristic X(M)>0 and that in the case of negative curvature X(M) (-1)^d >0. The Hopf product conjecture asks whether a positive curvature metric can exist on product manifolds like S^2 x S^2. By formulating curvature integral geometrically, these questions can be explored for finite simple graphs, where it leads to linear programming problems. In this more expository document we aim to explore also a bit of the history of the Hopf conjecture and mention some strategies of attacks which have been tried. We illustrate the new integral theoretic mu-curvature concept by proving that for every positive curvature manifold M there exists a mu-curvature K satisfying Gauss-Bonnet-Chern X(M)=\int_M K dV such that K is positive on an open set U of volume arbitrary close to the volume of M.