If $(M,g)$ is a smooth Riemannian manifold and $c : [a,b] \to M$ is a smooth embedded simple curve on $M$, it is always possible to choose locally a Riemannian metric $g\_0$ on $M$ for which $c$ is a geodesic for $g\_0$. As I understand, this can be done by pulling back a tubular neighborhood of $c$ to a disc bundle along $c$. There is always a flat Riemannian metric for which the $0$-section ($c$ itself) is a smooth geodesic. My question is: is it possible to locally realize $c$ as a geodesic to a metric such that the tube is positively curved? For instance, one of constant sectional curvature? I mean, can $g\_0$ be positively curved? If it helps, $M$ can be assumed to be a surface.



Discs bundles along a curve and positive curvature
It appears to me that your question does not involve the metric $g$ at all. Any connected embedded curve in a smooth $n$-manifold has a tubular neighborhood diffeomorphic to $T = I\times D$,
where $I$ is an interval or a circle and $D$ is the unit $(n-1)$-dimensional disk. You can map $T$ into the unit sphere such that $I\times \{0\}$ maps into a geodesic and and $T$ onto a tubular neighborhood of that geodesic. If $g\_0$ is the pullback of the standard metric on the sphere, then it has positive sectional curvature $1$.


