# Determining the metric through lengths of distance minimizing curves

Consider a two dimensional smooth Riemannian manifold $(M,g)$ with boundary $(N,h)$, where $h$ is the induced metric. For simplicity let $M$ be homotopic to a disk. Does specifying the length of distance-minimizing curves in $M$ between any two points $x,y\in N$ uniquely determine $g$ up to diffeomorphism $M\to M$ such that $N$ is fixed?