Local and Global Boundary Rigidity and the Geodesic X-ray Transform in the Normal Gauge

In this paper we analyze the local and global boundary rigidity problem for general Riemannian manifolds with boundary (M, g) whose boundary is strictly convex. We show that the boundary distance function, i.e. dg |∂M×∂M , known over suitable open sets of ∂M determines g in suitable corresponding open subsets of M , up to the natural diffeomorphism invariance of the problem. We also show that if there is a function on M with suitable convexity properties relative to g then dg |∂M×∂M determines g globally in the sense that if dg |∂M×∂M = dĝ |∂M×∂M then there is a diffeomorphism ψ fixing ∂M (pointwise) such that g = ψ∗ĝ. This global assumption is satisfied, for instance, for the distance function from a given point if the manifold has no focal points (from that point). We also consider the lens rigidity problem. The lens relation measures the point of exit from M and the direction of exit of geodesics issued from the boundary and the length of the geodesic. The lens rigidity problem is whether we can determine the metric up to isometry from the lens relation. We solve the lens rigidity problem under the same global assumption mentioned above. This shows, for instance, that manifolds with a strictly convex boundary and non-positive sectional curvature are lens rigid. The key tool is the analysis of the geodesic X-ray transform on 2-tensors, corresponding to a metric g, in the normal gauge, such as normal coordinates relative to a hypersurface, where one also needs to allow microlocal weights. This is handled by refining and extending our earlier results in the solenoidal gauge.