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 $(M,g)$. We show that distance function, i.e., $d_g|_{\partial M \times \partial M}$, known near a point $p\in M$ at which $\partial is strictly convex, determines $g$ in suitable neighborhood of $p$ $M$, up to natural diffeomorphism invariance problem. also consider closely related lens more formulation if not realized by unique minimizing geodesics. The relation measures direction exit from $M$ geodesics issued length geodesic. whether can determine metric isometry relation. solve under assumption there function on convexity properties relative $g$. This be considered as complete solution formulated first Herglotz 1905. prove semi-global results given data. shows, instance, simply connected convex boundaries are rigid sectional curvature non-positive or non-negative no focal points. key tool analysis geodesic X-ray transform 2-tensors, corresponding $g$, normal gauge, such coordinates hypersurface, where one needs allow weights. handled refining extending our earlier solenoidal gauge.