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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.:-)

         
          Wikipedia image: "A geodesic on a triaxial ellipsoid." Credit: Charles Karney.
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Joseph O'Rourke
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    It's worth noting that in the orbifold literature there are two distinct definitions of geodesics: locally length minimizing curves and curves that locally lift to geodesics. See for example the survey article Orbifolds and their Spectra by Carolyn Gordon. –  May 27 '17 at 01:41
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    What is the other characterization you found? – Mariano Suárez-Álvarez May 31 '17 at 22:28
  • @MarianoSuárez-Álvarez: Sorry, buried in a comment to Rodrigo, which I quote here: "The alternative characterization involves a "nowhere vanishing symmetric tensor field" along the curve $\gamma$, with an associated linear function on the tensor that satisfies a particular zero-equation. I do not understand this aspect except that it is (according to Sternberg) a characterization of geodesics." See Willie Wong's explanation in those comments. – Joseph O'Rourke Jun 01 '17 at 01:33
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    The question is specifically about Riemannian metrics, but you mix in some discussion of Einstein and GR. In the semi-Riemannian case, the definition in terms of parallel transport is not just more fundamental than the length-minimizing one -- the length-minimizing one simply doesn't work except in a very restricted sense. E.g., for +--- signature, timelike geodesics minimize length, but null and spacelike ones don't. (For a null geodesic, it's not even clear how to define extremization.) –  Jun 01 '17 at 14:26
  • @BenCrowell: Thanks for emphasizing that distinction. I'm so oriented myself toward the geometric viewpoint that I didn't see the bias toward length-minimizing (or length-preserving, à la Matt F). – Joseph O'Rourke Jun 02 '17 at 01:04

3 Answers3

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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.

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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.

Rodrigo A. Pérez
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  • The alternative characterization involves a "nowhere vanishing symmetric tensor field" along the curve $\gamma$, with an associated linear function on the tensor that satisfies a particular zero-equation. I do not understand this aspect except that it is (according to Sternberg) a characterization of geodesics. – Joseph O'Rourke May 27 '17 at 01:44
  • I did not emphasize the historical dimension in my question, so I appreciate that you did. – Joseph O'Rourke May 27 '17 at 01:58
  • @JosephO'Rourke Is that geodesics in general or time like geodesics in a Lorentzian manifold under assumptions? The EIH paper if I remember right is attacking the problem of whether a "test particle travels along a time-like geodesic"; the problem in GR being that the theory is nonlinear so you cannot just "add" test particles. – Willie Wong May 27 '17 at 02:00
  • @WillieWong: You are well ahead of me on the EIH paper. This is far from my expertise, and I defer to your recollections. – Joseph O'Rourke May 27 '17 at 02:02
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    @JosephO'Rourke: Sternberg's write up is much clearer than EIH's original. The characterization is indeed general. The characterization can be described as what happens if in the "calculus of variations" description of geodesic, instead of varying the curve by actually moving the curve, you vary the curve by "moving the manifold" instead (in the sense of one parameter family of local diffeomorphisms generated by a vector field of compact support). – Willie Wong May 27 '17 at 02:27
  • @WillieWong That sounds really interesting. Is something new gained when considering both variations at the same time? – Vít Tuček May 27 '17 at 07:30
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    @VítTuček I don't think EIH does both at the same time (the curve is fixed). In the context of (pseudo)Riemannian geometry, all three descriptions of the geodesics are transformable to each other with a few steps, so I don't really know whether anything is new. // On the other hand, in spite of my short interpretation above, the formulation is not as simple as "first variation of the length functional with respect to local diffeomorphisms"; there are a few subtleties involved to get an if and only if statement. – Willie Wong May 27 '17 at 13:05
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    Perhaps "primary" is in the eye of the beholder. I think for a geometer, "local minimizing" is the fundamental motivation. However, for a physicist, who is more used to working with non-positive-definite Hamiltonians, "stationary point of an energy functional" is perhaps the fundamental concept. And, given their past experience, the fact that this leads to constant acceleration curves is not a surprise. – Deane Yang May 27 '17 at 19:09
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    In the "géométricon", one of the adventures of Anselme Lanturlu by Jean-Pierre Petit, there is indeed a "handwaving description" of geodesics by autoparallelism. At least on surfaces, this works well: just try to follow a geodesic by gluing a very long piece of sticky tape on your surface. – Sebastian Goette May 28 '17 at 16:23
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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.

Cristi Stoica
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