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

Abstract

This paper considers the boundary rigidity problem for a compact convex Riemannian manifold ( M , g ) (M,g) with boundary ∂ M \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 ′ g’ on M M there is a C 3 , α C^{3,\alpha } -neighborhood of g g such that g 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 M ). More precisely, given any metric g ′ g’ in this neighborhood with the same boundary distance function there is diffeomorphism φ \varphi which is the identity on ∂ M \partial M such that g ′ = φ ∗ g g’=\varphi ^{*}g . There is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance-function.