Boundary rigidity and stability for generic simple metrics

We study the boundary rigidity problem for compact Riemannian manifolds with boundary : is the Riemannian metric uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function known for all boundary points and ? We prove in this paper local and global uniqueness and stability for the boundary rigidity problem for generic simple metrics. More specifically, we show that there exists a generic set of simple Riemannian metrics such that for any , any two Riemannian metrics in some neighborhood of having the same distance function, must be isometric. Similarly, there is a generic set of pairs of simple metrics with the same property. We also prove Hölder type stability estimates for this problem for metrics which are close to a given one in .