The definition of soft manifoldss now fixed. It is only fitting to call the old discrete manifolds hard manifolds. Unlike the later, soft manifolds can be multipied. They define a subring of the Shannon ring.
We report on some progress on analytic torsion A(G) for graphs. A(G) is a positive rational number attached to a network. We can compute it for contractible graphs or spheres.
A major open problem in Riemannian geometry is the classification of even dimensional positive curvature manifolds with symmetry. There is a reduction theory which produces a periodic system of elements. This picture has affinity with gauge bosons in physics.
About a new result in Riemannian geometry.
The Hopf conjectures were first formulated by Hopf in print in 1931. The sign conjecture claims that positive curvature compact Riemannian 2d-manifolds have positive Euler characteristic and that negative curvature compact Riemannian 2d-manifolds have Euler characteristic with sign (-1)d . The product conjecture claims there is no positive curvature metric …