= Boundary distance determines geodesic scattering
With the full boundary metric fixed, first variation gives $d_xd_g(w)=-\langle v_{\rm in},w\rangle$ and $d_yd_g(z)=\langle v_{\rm out},z\rangle$ for tangential endpoint variations. These covectors recover the tangential parts of both unit velocities. Their normal components follow from unit length and the inward/outward sign, so equal boundary distances give equal scattering relations. Without a fixed boundary metric, the bundle identification must first be gauged by a boundary-fixing diffeomorphism.
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