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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Gaussian curvatures of Lorentzian metrics on the plane and punctured planes


Author: Jiang Fan Li
Journal: Proc. Amer. Math. Soc. 108 (1990), 197-205
MSC: Primary 53C50; Secondary 35J60
MathSciNet review: 984805
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Abstract: We prove that every $ f \in {C^k}\left( {{R^2}} \right)$ is the Gaussian curvature of some $ {C^{k + 1}}$ -Lorentzian metric $ \left( {0 \leq k \leq \infty } \right)$. Let $ M$ denote the cylinder. We prove that every continuous function on $ M$ is the Gaussian curvature of some $ {C^1}$-Lorentzian metric. If $ f \in {C^k}\left( M \right)$ satisfies the condition (H) in the Lemma 2 below, then it is the curvature function of some $ {C^{k + 1}}$-Lorentzian metric. If $ f \in {C^k}\left( {{R^2}} \right)\left( {1 \leq k \leq \infty } \right)$ has compact support, then the Lorentzian metric can be made complete.


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DOI: https://doi.org/10.1090/S0002-9939-1990-0984805-8
Keywords: Curvature function, (complete) Lorentzian metric, characteristic line, (forward) complete geodesic
Article copyright: © Copyright 1990 American Mathematical Society