Geodesic exponential patches

We pick up a rope left last year about defining the exponential map in a simplicial complex. The geodesic flow does that for one dimensional directions. Already for defining two dimensional exponential sheets we can not just take "geodesics passing in a two dimensional directions" which does not make sense for many reasons, one being that we have no tangent space and that we would only have 3 geodesics and not a 2-dimenaional manifold passing through the position we want to define it. THere is some mathematica code here: https://www.quantumcalculus.org/exponential-patch which illustrates the story so far. We will continue this next week and show how to get the entire manifold. The blog entry also explains a bit more why going from the Frame bundle to the Grassmannian bundle is a crucial step. Pictures were taken last sunday march 29th on the Mystic lake