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If I understand correctly, a geodesic between two points $a$ and $b$ is the "most direct" path from $a$ to $b$. Geodesics on a plane are straight lines, geodesics on a sphere are great circle arcs. Geodesics can be defined on any Riemannian manifold (right?).

If I've got that roughly correct, then what might be the "opposite" of a geodesic? And can a unique "opposite" be defined?

What about this definition: let a cisedoeg (the opposite of a geodesic) be a curve that connects $a$ and $b$ but that nowhere intersects the geodesic between $a$ and $b$. Also let the tangents to the cisedoeg all be parallel.

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Geodesics come from trying to minimize the energy functional, and so perhaps the opposite of geodesics come from attempting to maximize the energy functional. It seems unlikely that in total generality such maxima will exist though. –  Alex Youcis Dec 15 '12 at 18:42
@AlexYoucis: Actually, geodesics are curves of stationary (not necessarily maximal or minimal) length. –  Henning Makholm Dec 15 '12 at 18:46
For what it's worth, I seem to recall that on Lorentz manifolds, geodesics locally maximize, not minimize, length. –  Neal Dec 15 '12 at 19:41
@Neal: Timelike geodesics in Lorentzian manifolds maximize length; spacelike ones neither maximize nor minimize the length (you can make a longer neighboring curve by varying it in a spacelike direction, or a shorter neighboring curve by varying it in timelike direction). –  Henning Makholm Dec 15 '12 at 21:41

2 Answers 2

up vote 6 down vote accepted

No, a differentiable manifold is not enough to have geodesics. You also need a metric on the manifold, which makes it a Riemannian (or pseudo-Riemannian) manifold.

Your definition involving

Also let the tangents to the cisedoeg all be parallel.

doesn't really make sense because manifolds (Riemannian or otherwise) do not come with a canonical identification of the tangent spaces that you can use to define whether two tangents at different points are parallel.

In a Riemannian manifold you can require that the tangents to the curve all move into each other when "parallel transported" along the curve. However, this turns out to be an alternative characterization of geodesics, not some kind of anti-geodesic.

I suspect the source of your problem is that your basic premise

If I understand correctly, a geodesic between two points a and b is the "most direct" path from a to b.

is not completely correct. A geodesic is a curve that is locally the shortest curve between two points on it -- where "locally" means that it only means to be shortest between two suffiently close points on the geodesic.

In particular a great-circle arc on a sphere that is longer than 180° is still a geodesic. (An entire great circle, traversed again and again as the parameter increases to $\infty$, is a geodesic).

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Thanks for the education! :D I gained a better understanding of what a geodesic is and I learned the difference between a differential and Riemannian manifold. –  John Berryman Dec 15 '12 at 18:49
@Henning, just to complement your answer, besides the metric and the differential manifold structures, one also needs a connection, since it's a separate structure and is used to define geodesics. –  Iberê Kuntz Apr 26 at 15:53
@IberêKuntz: I'm assuming that when nothing else is specified, the Levi-Civita connection (which is derived from the metric) is to be used. –  Henning Makholm Apr 26 at 16:04

By "opposite", do you mean like lines that are orthogonal to every geodesic or something? Because I'm just thinking about $\Bbb R^2$ and no such cisedoeg exists : if all tangents are parallel, you have a straight line ; but the only straight line that connects $a$ and $b$ in the plane is the straight line passing through them, i.e. the geodesic.

I don't think your definition makes sense at all, nor do I see what you're trying to build. Perhaps you should give us more feeling about it.

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On a sphere, I'm referring to the cisedeog as the longer arc of the Great Circle connecting $a$ and $b$. This is a trivial example (and as Henning pointed out, I had a misunderstanding of what a geodesic really is). However I do think that it's an interesting question as to whether or not a cisedeog exists on an arbitrary Riemannian manifold. Consider the case of a cisedeog connecting two points on a torus. –  John Berryman Dec 15 '12 at 18:55
On a flat torus, I believe the set of all geodesics from $a$ and $b$ is dense. –  Hurkyl Dec 15 '12 at 19:12
@Hurkyl really? How could that be? (I'm having trouble visualizing that.) I would be interested you have any references. –  John Berryman Dec 15 '12 at 19:18
@John: Imagine the torus as the unit square (where, if you go off one edge, you reappear on the opposite edge, like in a video game). Any line starting from the origin with odd integral slope is a geodesic that will pass through the origin as well as the midpoint of the square. The greater the slope, the more times it will wind around the torus as it goes from one point to the other. –  Hurkyl Dec 15 '12 at 19:21
@John : The curve you are describing in your comment is a geodesic, it's just not the geodesic from $a$ to $b$. A curve is not said to be geodesic with respect to points ; it's a property of the curve itself. Being the shortest path between two points is a particular case of geodesic curves. –  Patrick Da Silva Dec 15 '12 at 19:58

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