Before revision:  We focus on the pragmatic and calculable results and avoid the use of tensor notation to appeal to a broader audience . We have already used these derivations for important advances in our own work (for instance, geometric fabrics); it is our hope that they will contribute to an increased understanding of these ideas within the robotics community and inspire future research into applications of generalized nonlinear, Finsler, and even Riemannian geometries .
After revision:  We focus on the pragmatic and calculable results and avoid the use of tensor notation to appeal to a broader audience and emphasizing geometric path consistency over ideas around connections and curvature. It is our hope that they will contribute to an increased understanding of these ideas within the robotics community and inspire future research into applications of generalized nonlinear, Finsler, and even Riemannian geometries .