While the apparent straightness of a curve depends on the choice of coordinates, there's also an invariant notion defined in terms of the connection. This affe

While the apparent straightness of a curve depends on the choice of coordinates, there's also an invariant notion defined in terms of the connection. This affects our understanding of forces: Accelerations live in the double tangent bundle, which gets split into a horizontal and vertical part by the connection. The horizontal part is the contribution by inertia, yielding pseudo-forces (including gravity according to GR). The vertical part is due to non-inertial forces.