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Boundary and Lens Rigidity of
Lorentzian Surfaces

Authors: Lars Andersson, Mattias Dahl and Ralph Howard
Journal: Trans. Amer. Math. Soc. 348 (1996), 2307-2329
MSC (1991): Primary 53C50
MathSciNet review: 1363008
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Abstract: Let $g$ be a Lorentzian metric on the plane $\r ^2$ that agrees with the standard metric $g_0=-dx^2+dy^2$ outside a compact set and so that there are no conjugate points along any time-like geodesic of $(\r ^2,g)$. Then $(\r ^2,g)$ and $(\r ^2,g_0)$ 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.

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Additional Information

Lars Andersson
Affiliation: Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden

Mattias Dahl
Affiliation: Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden

Ralph Howard
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

Received by editor(s): January 18, 1995
Additional Notes: Lars Andersson supported in part by the Swedish Natural Sciences Research Council (SNSRC), contract no. F-FU 4873-307. Mattias Dahl supported in part by the Wallenberg foundation. Ralph Howard supported in part by the SNSRC, contract no. R-RA 4873-306, the Swedish Academy of Sciences and the Crafoord foundation.
Article copyright: © Copyright 1996 American Mathematical Society