Integral geometric Hopf conjectures
Abstract
The Hopf sign conjecture states that a compact Riemannian 2dmanifold 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 mucurvature concept by proving that for every positive curvature manifold M there exists a mucurvature K satisfying GaussBonnetChern 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 DOI:
 10.48550/arXiv.2001.01398
 arXiv:
 arXiv:2001.01398
 Bibcode:
 2020arXiv200101398K
 Keywords:

 Mathematics  Differential Geometry;
 05C10;
 57M15;
 53Axx
 EPrint:
 20 pages, 2 figures