# Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge

## TL;DR

This paper investigates boundary and lens rigidity problems for Riemannian manifolds, establishing conditions under which the metric can be uniquely determined from boundary data, and extends analysis of the geodesic X-ray transform in the normal gauge.

## Contribution

It provides new results on local and global boundary and lens rigidity, solving the lens rigidity problem under convexity assumptions, and advances the analysis of the geodesic X-ray transform in the normal gauge.

## Key findings

- Boundary distance function determines the metric near convex boundary points.
- Lens rigidity is achieved under convexity and curvature conditions.
- Extended analysis of the geodesic X-ray transform in the normal gauge.

## Abstract

In this paper we analyze the local and global boundary rigidity problem for general Riemannian manifolds with boundary $(M,g)$. We show that the boundary distance function, i.e., $d_g|_{\partial M\times\partial M}$, known near a point $p\in \partial M$ at which $\partial M$ is strictly convex, determines $g$ in a suitable neighborhood of $p$ in $M$, up to the natural diffeomorphism invariance of the problem. We also consider the closely related lens rigidity problem which is a more natural formulation if the boundary distance is not realized by unique minimizing geodesics. The lens relation measures the point and the direction of exit from $M$ 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 assumption that there is a function on $M$ with suitable convexity properties relative to $g$. This can be considered as a complete solution of a problem formulated first by Herglotz in 1905. We also prove a semi-global results given semi-global data. This shows, for instance, that simply connected manifolds with strictly convex boundaries are lens rigid if the sectional curvature is non-positive or non-negative or if there are no focal points. 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 weights. This is handled by refining and extending our earlier results in the solenoidal gauge.