A closed geodesic in a q-manifold is a q-manifold C with boundary dC. This boundary dC is a circle bundle. It can be for example. But it can also be a non-trivial bundle. Note that everything is purely combinatorial and pretty small. For the Moebius strip for example C is generated by the geodesic {{1, 2, 3}, {2, 3, 4}, {3, 4, 5}, {4, 5, 6}, {5, 6, 7}, {6, 7, 1}, {7, 1, 2}}. For the cylinder just make the geodesic {{1, 2, 3}, {2, 3, 4}, {3, 4, 5}, {4, 5, 6}, {5, 6, 7}, {6, 7, 8}, {7, 8, 1}, {8, 1, 2}}. The case q=2 is not yet that interesting. In the case q=3, C is either a solid torus or a solid Klein bottle. Interesting is the case q=4, were we get 3-manifolds. Remember that we have defined sectional curvature for any manifold. Now we measure: there are discrete manifolds which have all sectional curvatures positive.
Update April 25, 2025: This clashes with the continuum, where we have exceptional 2-dimensional directions (elements in the Grassmannian Gr(q,2))) which lead to zero sectional curvature. This example is also a bit of a warning for a discrete approach to the Hopf problem to find positive curvature metric on , one of my personal favorite problems I pursued for many years (under the assumption of course that there is one). The “no-impediment” principles discussed in this presentation would suggest that there must be a positive metric metric on the Hopf manifold because “if there is no impediment or obstruction to construct one, there must be one”. I had presented this principle along other non-provable meta laws like the law of small numbers. Such principles are not laws but they are good guidelines. I would expect soon to find discrete examples of for which all sectional curvatures are positive (in the sense described since last winter). Again, also in the Hopf manifold case, the Grassmannian manifold of 2 dimensional directions in a 4 dimensional space for which the classical curvatures are zero is of measure zero.
Update April 25, 2025: This is a big surprise as I had really hoped for a nice sphere theorem with the notion of curvature developed under reasonable pinching conditions. Not all is lost as is not simply connected. Still, I have had the impression that positive curvature would allow for an easy Myers-Bonnet type bounds on the size of the manifold. The example just discovered shatters such a hope. One reason why I was stumped with this example is that I somehow in the back of my head always assumed that if we construct the geodesic sheet through a bone (a simplex) in the manifold, that the geodesic would stay on that sheet. But if you look at the construction (I had even posted code a few weeks ago to get these sheets), this is obviously not the case, we move on the geodesic and the bone when moving on the wheet. If the geodesics would remain on geodesic sheets, we would immediately give upper bounds on the diameter of the manifold. In the continuum, a geodesic sheet (classically the image of a 2-dimensional plane in the tangent space under the exponential map at x ) can be very complicated, even dense if the geodesic flow is ergodic. In the discrete, the geodesic sheets are nice closed manifolds in the frame bundle of the manifold (which of course is a finite set of elements). In the positive curvature case, of course, all these geodesic sheets are 2-spheres or 2-dimensional projective planes. By a discrete Synge argument (I always called it “Geomag argument” in the didvtryr because of the toy Geomag I played with a lot), it must in even dimensional orientable case be a 2-sphere. In the quantized discrete case, the global geodesics through a simplex containing a bone is not on the geodesic sheet of the bone in general. We have the strange situation that we have manifolds of the form where the factor can be arbitrary long such that there is a bound on the diameter of the geodesic sheets (a consequence of the positive curvature property and because of Gauss-Bonnet applied to the geodesic sheet) but no bound on the length of closed periodic orbits of the geodesic flow. [P.S. It had been a major punch line of our definition of sectional curvature that it satisfies Gauss Bonnet on the geodesic sheet or for 2-manifolds in general. Doing averages of Eberhard curvatures would not work unless the vertex degrees would be constant, but who wants to restrict discrete manifolds to be regular graphs? – It would be topic for graph theory but not for an attempt to emulate classical differential geometry overall.]
Update April 23, 2025: Here is some code. It allows to explore the cohomology of the complexes discussed. Also includes the infamous 5 line code for computing the cohomology of a simplicial complex. We just use it for the boundary complex in 3 dimensions which is either a torus or a Klein bottle.
