Given a finite collection of embedded $C^\infty$ curves which pass through the origin in $\mathbb{R}^2$ with different tangent directions and never again intersect, is there a clean way of prescribing a Riemmannian metric whose geodesics include those curves?
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Sorry, I misread your question and answered a local version instead. I'll leave it here (under the horizonal line) in case it turns out to be useful.
If your finite set of curves is rectifiable (can be made into lines by a diffeomorphism of the plane), then do that and then take the Euclidean metric. So the question: how wild are your curves?
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The answer is no. Being the geodesics of some Riemannian metric even in two dimensions is a quite restrictive condition. Even if you weaken this by only asking the curves to be the geodesics of some affine connection, it is still quite restrictive. Here is my favorite example of how rigid is this situation:
Theorem (A. Khovanskii). If (the pieces of) six circles on the plane that intersect at the origin are the geodesics of an affine connection defined on a neighbourhood of the origin, then all six circles must meet again at another common point.
If my memory serves me right this is in this paper. Maybe there is a little thing to add to the way he formulates his result in order to get to the theorem I stated: the exponential map of a $C^2$ connection is $C^2$ at the zero section. This allows you to use it to "rectify" the circles.
On the other hand, Finsler metrics are much more flexible for this sort of thing (and their exponential maps are just $C^1$ on the zero section). You can take a look at this paper and its reference for this sort of thing.
One more comment: even if your curves are the geodesics of an affine connection, they still have some way to go before being the geodesics of some Riemannian metric. This was studied here by Bryant, Dunajski, and Eastwood.
