Riemannian geometry is related to general relativity, probability theory is related to quantum mechanics. Length enters in GR in the form of geodesics, paths of mass points and curvature, the deformation of space through mass. Probability enters in quantum mechanics by seeing solutions of wave or Schrödinger equations in terms …
A discrete 2-manifold [PDF] is a finite simple graph for which all unit spheres have d(v)=4 or more vertices. The smallest example is which is the octahedron graph and where every unit sphere is a circular graph with 4 elements. The Eberhard curvature is very rigid: the Mickey Mouse theorem …
Curvature is a local notion in a finite geometry that adds up to a topological invariant. This is Gauss-Bonnet. I’m only really interested in curvatures that satisfy this exactly. This does not exclude sectional curvature, the latest venture because sectional curvature integrated over a geodesic sheet is the Euler characteristic …
We continue the quest to define a sectional curvature for q-manifolds. A good notion should produce classical theorems like that if sufficiently pinched manifolds are spheres. Asking all embedded wheel graphs to have positive curvature was much too rigid and produced only spheres, so small that I called this the …
A 2-manifold with boundary is a finite simple graph for which every unit sphere is a circular graph with 4 or more nodes or a path graph with 3 more nodes. The boundary is the set of vertices for which the unit sphere is a path graph, the interior is …
This is a presentation from Saturday, July 13, 2024. Curvatures are usually located on the zero dimensional part of space. I look here at curvature located on one or two dimensional parts of space. In the special case of a triangulation of a 2-dimensional surface, where the usual curvature is …
Here are three catchy open problems in differential geometry. As with any problem, we can look how to formulate discrete versions. The first problem is whether a positive curvature 2d manifold has positive Euler characteristic, the second is whether there is a positive curvature metric on and the third is …
The Curvature of graphs multplies under the Shannon product (strong product).
The Mickey mouse theorem assures that a connected positive curvature graph of positive dimension is a sphere.
The f-function of a graph minus 1 is the sum of the antiderivatives of the f-function anti-derivatives evaluated on the unit spheres.
