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?
In 2005 Pestov and Uhlmann showed that distances between boundary points determine the Riemannian metric up to diffeomorphism:
Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math. 161 (2005), no. 2, 1093--1110; MR2153407 (2006c:53038).