Geodesic scattering relation ($\alpha (x,v)=(\gamma (\tau ),\dot{\gamma }(\tau ))$)

For an inward unit boundary vector, the geodesic scattering relation returns the next boundary point and its outgoing unit velocity. On outward vectors it can be extended using the first backward exit, giving an involution on the full boundary unit bundle, with tangential vectors fixed. Exit time is separate lens data; on a simple manifold it equals boundary distance between the endpoint pair.

With the full boundary metric fixed, first variation gives $d_x{d}_g(w)=-⟨v_{\mathrm{in}},w⟩$ and $d_y{d}_g(z)=⟨v_{\mathrm{out}},z⟩$ 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.