Boundary and Lens Rigidity of Lorentzian Surfaces

LARS ANDERSSON, MATTIAS DAHL, AND RALPH HOWARD Abstract. Let g be a Lorentzian metric on the plane R2 that agrees with the standard metric g0 = −dx + dy outside a compact set and so that there are no conjugate points along any time-like geodesic of (R2 , g). Then (R2 , g) and (R2 , g0) are isometric. Further, if (M, g) and (M∗, g∗) are two dimensional compact time oriented Lorentzian manifolds with space–like boundaries and so that all time-like geodesics of (M, g) maximize the distances between their points and (M, g) and (M∗, g∗) are “boundary isometric” then there is a conformal diffeomorphism between (M, g) and (M∗, g∗) and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.