Local boundary rigidity of a compact Riemannian manifold with curvature bounded above

This paper considers the boundary rigidity problem for a compact convex Riemannian manifold with boundary whose curvature satisfies a general upper bound condition. This includes all nonpositively curved manifolds and all sufficiently small convex domains on any given Riemannian manifold. It is shown that in the space of metrics on there is a -neighborhood of such that is the unique metric with the given boundary distance-function (i.e. the function that assigns to any pair of boundary points their distance -- as measured in ). More precisely, given any metric in this neighborhood with the same boundary distance function there is diffeomorphism which is the identity on such that . There is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance-function.