Geodesic for a linear connection
Consider a smooth curve . Let denote the connection along induced by , and consider the transport along for the connection . Then, we say that is a geodesic for if, under that transport, the tangent vector at gets transported, at time , to the tangent vector at .
Equivalently, we say that is a geodesic if:
(with the derivative interpreted as a suitable one-sided derivative at the endpoints).