Primary definition of a geodesic

Question

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.

1. A geodesic has acceleration zero, i.e., it is self-parallel.

2. 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.:-)

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          ^{Wikipedia image: "A geodesic on a triaxial ellipsoid." Credit: Charles Karney.}

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Answer 1

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.

1. 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.

2. 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.

Answer 2

Herbert Busemann defined a geodesic as "a locally isometric map of the real axis" in a metric space. So on his point of view, a geodesic is not primarily length-minimizing but length-preserving.

In his 1955 book, The Geometry of Geodesics, he used this definition (p. 32), and a few other properties to prove that:

• (7.9) A geodesic exists through any two points.
• (8.12) If geodesics converge pointwise, they converge to a geodesic, uniformly on bounded intervals.
• (31.2) If the geodesic through any two points is unique, either all geodesics are isometric images of the real line or all are great circles of the same length.

This approach to metric geometry indeed allowed him to generalize Riemannian geometry. One of his recurrent themes was that, by comparison with coordinatized Finsler geometry, these definitions made it easier to generalize geometry well.

Answer 3

I have never seen the term "geodesic" explained in a hand-waving fashion in any other way than as a shortest path curve. From this (meager) evidence, I am willing to assume that, historically, the local distance minimization was the original motivating idea. After all, what is a primary activity of surveyors when they do their geodetic measurements? I'm willing to believe that is the etymological origin of the word geodesic.

It would also not surprise me if it took some time for 19th Century people to realize that "shortest distance" must be interpreted locally.

By the way, it would be great if you could add the alternative definition you found, just for completeness of your question, together with an explanation of the nice picture.