Boundary and Lens Rigidity of Lorentzian Surfaces
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- by Lars Andersson, Mattias Dahl and Ralph Howard PDF
- Trans. Amer. Math. Soc. 348 (1996), 2307-2329 Request permission
Abstract:
Let $g$ be a Lorentzian metric on the plane $\mathbb {R}^2$ 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 $(\mathbb {R}^2,g)$. Then $(\mathbb {R}^2,g)$ and $(\mathbb {R}^2,g_0)$ 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.References
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Additional Information
- Lars Andersson
- Affiliation: Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
- Email: larsa@math.kth.se
- Mattias Dahl
- Affiliation: Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
- Email: dahl@math.kth.se
- Ralph Howard
- Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
- MR Author ID: 88825
- Email: howard@math.sc.edu
- 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.
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 2307-2329
- MSC (1991): Primary 53C50
- DOI: https://doi.org/10.1090/S0002-9947-96-01688-1
- MathSciNet review: 1363008