I am wondering if there is a sense in which one of these definitions for a geodesic on a smooth Riemannian manifold is primary to the other.
A geodesic has acceleration zero, i.e., it is self-parallel.
A geodesic is locally length-minimizing, i.e., it is stationary relative to variations in arc length.
I encountered another characterization—in the Einstein, Infeld, Hoffmann paper, "Gravitational Equations and the Problem of Motion," cited in Shlomo Sternberg's book on Curvature in Mathematics and Physics—which seems far from primary, and made me wonder if there is any natural priority between the two definitions above. Of course, each can be mathematically derived from the other, but I am thinking that, e.g., one might more easily generalize than the other, or somehow be more pure to the spirit of Riemannian geometry.
Image below included just for aesthetic appreciation.:-)
Wikipedia image: "A geodesic on a triaxial ellipsoid." Credit: Charles Karney.