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 $(M,g)$ with boundary $\partial M$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 $g'$ on $M$ there is a $C^{3,\alpha }$-neighborhood of $g$ such that $g$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 $M$). More precisely, given any metric $g'$ in this neighborhood with the same boundary distance function there is diffeomorphism $\varphi$which is the identity on $\partial M$such that $g'=\varphi ^{*}g$. There is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance-function.