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

摘要

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