The tangent bundle exponential map and locally autoparallel coordinates for general connections with application to Finslerian geometries

We construct a tangent bundle exponential map and locally autoparallel coordinates for geometries based on a general connection on the tangent bundle of a manifold. As concrete application we use these new coordinates for Finslerian geometries and obtain Finslerian geodesic coordinates. They generalise normal coordinates known from metric geometry to Finsler geometric manifolds and it turns out that they are identical to the Douglas-Thomas normal coordinates introduced earlier. We expand the fundamental geometry function of a Finsler spacetime in these new coordinates and find that it is constant to quadratic order. The quadratic order term comes with the non-linear curvature of the manifold. From physics these coordinates may be interpretation as the realisation of an Einstein elevator in Finslerian spacetime geometries.