Title: Local and global boundary rigidity and 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 $(M,g)$ whose boundary is strictly
convex. We show that the boundary distance function, i.e., $d_g|_{\partial
M\times\partial M}$, known over suitable open sets of $\partial M$ 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
$d_g|_{\partial M\times\partial M}$ determines $g$ globally in the sense that
if $d_g|_{\partial M\times\partial M}=d_{\tilde g}|_{\partial M\times \partial
M}$ then there is a diffeomorphism $\psi$ fixing $\partial M$ (pointwise) such
that $g=\psi^*\tilde g$. 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.