I'd like to know how the parallel transportation behaves in non-Levi-Civita connections and how does one realize it formally.
I know that parallel transportation along some piece-smooth curve is defined through moving by geodesics: say, in case of conformal connection, one transports a vector along a short geodesic close to a piece of the given curve preserving the angle at each moment of moving, so, for example moving by a horizontal line in the flat realization of the Lobachevsky plane (where geodesics are semicircles and vertical lines) one gets that a vector rotates uniformly.
But how does one understand the parallel transport in general (of a general connection given on a principal bundle)? So, are there any axioms of the parallel transport except being an isomorphism between model spaces in close points on a smooth manifold and "additivity: moving along one piece of a curve and then the second" and "inverse: going in both sides" axioms?