Positive Curvature Definition

After talking about this on Saturday, I tried some other things (this is a perfect topic to think about before getting to sleep or even contemplate about while sleeping). First of all, we can extend how to evolve the geodesic flow given a triangle t=(a,b,c) in a q-simplex x of the q-manifold G. The points in the triangle defines walls (which are (q-1) simplices) x-{a},x-{b},x-{c}) through which we can evolve the geodesic flow. While the simplices directly adjacent are independent the orientation, the continuation does depend on orientation. A substantial further enlargement of the discrete Grassmannian is to allow both the forward as well as the backward geodesic flow. These are different things as we chose a rotation of the simplex when moving forward. Given a triangle t=(a,b,c) within a simplex x, we have now 6 different directions to go: forward and backwards through each of the edges. We can now chose three of these 6 directions to form a geodesic sheet.

I tried that early Sunday morning and indeed the curvature spectrum of all possible planes through a point (defining each a geodesic sheet through the point) gets larger. Still there are also with this no zero curvature planes yet for the boundary manifolds of geodesics. We must note however that we deal with very small simplicial complexes here that are not Whitney complexes of graphs. Indeed, the construction of obtained by taking the boundary of a geodesic tube in a 4 manifold produces very small manifolds. We match the smallest simplicial complex constructions in the Manifold library of Frank Lutz in the case of the 3-manifold and can get like this small versions of as well as twisted versions of this in arbitrary dimensions. In the case of 2-manifolds, we get a torus with 14 maximal simplices (facets), for we get 44 facets, which matches the best known implementation of Frank H Lutz (who passed away in 2023 by the way). For the 4-manifold we get an implementation with 70 facets and for the 5-manifold we have an implementation with only 99 facets. Of course, any graph implementation produces much more simplices. I had worked with a student (James Chen) last year about discrete systolic geometry which essentially is a hunt for small manifolds. James found a 2-torus with 24 triangles and systole 4 leading to the systolic constant 16/24=2/3. We believe this is the smallest possible 2-torus. In the discrete systolic geometry approach which James Chen had followed, we look at manifolds coming from graphs. The systole can not be 3 for example. Again there is a culture clash here as a graph theorist would say the girth is the systole (which is the attitude not to count the faces).

The latest I tried is take a Barycentric refinement of such a manifold and indeed, now we see zero curvature sections. Is this the right approach? Barycentric refinements are a bit tricky with respect to curvature because Barycentric refinements do not preserve positive curvature manifolds. This is even the case for soft Barycentric refinements. So, we could work with the attempt that a q-manifold is defined to have positive curvature if both G and its Barycentric refinement have positive curvature. This is slightly stronger and might be strong enough to establish the three points we want to have satisfied: A) we can estimate a finite diameter of positive curvature manfolds. B) we get the same positive curvature manifolds than in the continnujm C) we can prove a sphere theorem. An other possibility which does not involve Barycentric refinement is to ask that the manifold be defined as a graph (meaning that it is the simplicial complex of a graph). Of course, every Barycentric refinement by definition has this property.

[I had worked in the first couple of years of this project (2010-2015) almost exclusively in graph frame works but moved slowly to a slightly more general simplicial complex language especially after realizing that most mathematicians did not understand what I was doing early on. This is related to the very strange attitudeof seeing graphs as one dimensional simplicial complexes. Not coming from graph theory myself, I always associated with a graph its Whitney complex, an octahedron for example is a 2 dimensional sphere with Euler characteristic 6-12+8=2 rather than 6-12=-6, as somebody would define it when looking at a graph as a “one dimensional simplicial complex”. Higher dimensional structures would for topological graph theorists only come in when doing realizations on actual manifolds, but then one involves classical topology using real manifolds, like surface and leaves combinatorics. Or then one would look at strange constructs like “multigraphs”, again a bit silly because a multigraph is just a set of sets which is a too general object if one does not add additional structure like assuming it to be closed under taking non-empty subsets or then look at delta sets. As pointed out several times already on these pages, sets of sets do not have a nice calculus while delta sets and in particular simplicial complexes do have a nice calculus. Starting with a graph is still the most intuitive notion for me as from my own experience of learning mathematics, I had no difficulty learning what a graph is in high school but had difficulty to learn calculus in high school. Sets of sets became natural in college, when seeing both topology and measure theory rigorously developed without any magicwand, which is usually involved if you look at calculus the first time; the magic wand would only really go away in “real analysis”. Indeed, we can still to this day wonder about whether classical calculus really is well defined as it is founded on an axiom system for which we do not know (and never will know) its consistency. Back to the story, So, maybe we need to assume that the manifold under consideration is a Whitney complex (which is satisfied if we look at the Barycentric refinement of the complex). At least we do not have a counter examples to A),B),C) yet any more. ]

By the way, the definition on the left has a typo . Things are maybe a bit more intuitive in a graph setting, where is a (q-1) sphere, a graph.