dg.differential geometry - Prescribing a Riemannian metric along a given geodesic

Let $gamma : [0,1] to mathbb R^2$ be a finite $C^2$-curve in the plane which does not intersect itself. Let $p(z)$ be a second-degree polynomial in $z in mathbb R^2$. Can we construct a Riemannian metric $g$ along $gamma$ such that

• $gamma$ is a geodesic of $g$,



• $gamma$ has length 1, and



• $p(z)$ is the 2-jet of $g$ at $gamma(0)$? (i.e. this prescribes $g$ and its first and second derivatives at $gamma(0)$) Edit: As per Sergei's comment, assume that $p$ is chosen so that $gamma$ does in fact solve the geodesic equation.

I think so, and my sketch of an argument follows the proof of existence of Fermi coordinates in reverse. I haven't worked through this in detail yet, though, because I'm more concerned about the next question:

Let $gamma$ and $eta$ both be finite curves $[0,1] to mathbb R^2$ which do not intersect themselves, and such that $gamma(0) = eta(0)$ and $gamma(1) = eta(1)$ with no other intersections (i.e. $gamma cup eta$ is a piecewise, simple, closed $C^2$-curve in the plane). Can we construct a metric $g$ as above? Note that if so, $gamma(0)$ and $gamma(1)$ will be conjugate points along $gamma$.

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