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

Understand

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.