TITLE: Primary definition of a geodesic
QUESTION [12 upvotes]: 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.

REPLY [5 votes]: I think that depending on what is the most fundamental structure you consider in your Riemannian manifold, both answers can be true. Let me explain.

In Euclidean geometry, one can consider the affine structure as more fundamental than the metric structure, so in this case the lines are more fundamental than lengths and angles. In general, this would mean that one considers the connection as more fundamental, even if the manifold happens to be Riemannian and the connection happens to be metric. So the answer would be 1.
But if the Riemannian manifold comes primarily from a metric space, then one can define curves and lengths of these curves, even if it happens that the topological manifold is also differentiable, and there is a metric tensor giving the same distances as the metric structure. In this case the connection appears only because of the metric, as its associated Levi-Civita connection, so the notion of parallel transport and of self-parallelism are secondary. So in this case the answer is 2.

I think there are cases in which the path leading to the Riemannian manifold, the order of the layers of different structures, really matters. One example can be in General Relativity - there are approaches where the causal structure is seen as more fundamental than the metric tensor, which allows the recovery of the metric up to the volume form, but even with only the causal structure much of the geometric and physical properties make sense. Of course we are talking here about the causal structure, which seems more basic than the affine structure, since we only know the null geodesics. I can expect that in general, both in applications of geometry and in abstract mathematical problems, spaces of parameters arise, and they may be endowed with one structure or another at the most basic level, and extra information allows adding the other structures on top of these.