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.
