  For a compact Riemannian manifold with boundary, we want to find the metric
structure from knowledge of distances between boundary points. This is called
the "boundary rigidity problem". If the boundary is not concave, which means
locally not all shortest paths lie entirely in the boundary, then we are able
to find the Taylor series of the metric tensor ($C^\infty$ jet) at the
boundary. In this paper we give a new reconstruction procedure for the jet at
non-concave points on the boundary using the localized boundary distance
function. A closely related problem is the "lens rigidity problem", which asks
whether the lens data determine metric structure uniquely. Lens data include
information of boundary distance function, the lengths of all geodesics, and
the locations and directions where geodesics hit the boundary. We give the
first examples that show that lens data cannot uniquely determine the jet. The
example include two manifolds with the same boundary and the same lens data,
but different $C^2$ jets. With some additional careful work, we can find
examples with different $C^1$ jets, which means the boundaries in the two
lens-equivalent manifolds have different second fundamental forms.
