Is there an "opposite" of a geodesic?

Question

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.

Answer 1

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.

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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).

Answer 2

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.