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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Conformally homeomorphic Lorentz surfaces need not be conformally diffeomorphic


Authors: Robert W. Smyth and Tilla Weinstein
Journal: Proc. Amer. Math. Soc. 123 (1995), 3499-3506
MSC: Primary 53C50; Secondary 53A30
MathSciNet review: 1273526
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Abstract: A Lorentz surface $ \mathcal{L}$ is an ordered pair (S, [h]) where S is an oriented $ {C^\infty }$ 2-manifold and [h] the set of all $ {C^\infty }$ metrics conformally equivalent to a fixed $ {C^\infty }$ Lorentzian metric h on S. (Thus Lorentz surfaces are the indefinite metric analogs of Riemann surfaces.) This paper describes subsets of the Minkowski 2-plane which are conformally homeomorphic, but not even $ {C^1}$ conformally diffeomorphic. It also describes subsets of the Minkowski 2-plane which are $ {C^j}$ but not $ {C^{j + 1}}$ conformally diffeomorphic for any fixed $ j = 1,2, \ldots $. Finally, the paper describes a Lorentz surface conformally homeomorphic to a subset of the Minkowski 2-plane, but not $ {C^1}$ conformally diffeomorphic to any subset of the Minkowski 2-plane.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1273526-7
PII: S 0002-9939(1995)1273526-7
Article copyright: © Copyright 1995 American Mathematical Society