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The space of Lorentzian flat tori in anti-de Sitter $ 3$-space


Authors: María A. León-Guzmán, Pablo Mira and José A. Pastor
Journal: Trans. Amer. Math. Soc. 363 (2011), 6549-6573
MSC (2010): Primary 53C42, 53C50
DOI: https://doi.org/10.1090/S0002-9947-2011-05324-9
Published electronically: July 22, 2011
MathSciNet review: 2833568
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Abstract: We describe the space of isometric immersions from the Lorentz plane $ \mathbb{L}^2$ into the anti-de Sitter $ 3$-space $ \mathbb{H}_1^3$, and solve several open problems in this context raised by M. Dajczer and K. Nomizu in 1981. We also obtain from the above result a description of the space of Lorentzian flat tori isometrically immersed in $ \mathbb{H}_1^3$ in terms of pairs of closed curves with wave-front singularities in the hyperbolic plane $ \mathbb{H}^2$ satisfying some compatibility conditions.


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Additional Information

María A. León-Guzmán
Affiliation: Departamento de Matemáticas, Universidad de Murcia, Spain
Email: maleong@um.es

Pablo Mira
Affiliation: Departamento de Matemática Aplicada y Estadĭstica, Universidad Politécnica de Cartagena, E-30203 Cartagena, Murcia, Spain
Email: pablo.mira@upct.es

José A. Pastor
Affiliation: Departamento de Matemáticas, Universidad de Murcia, Spain
Email: josepastor@um.es

DOI: https://doi.org/10.1090/S0002-9947-2011-05324-9
Keywords: Timelike flat surfaces, Lorentzian flat tori, isometric immersions, anti-de Sitter space
Received by editor(s): May 25, 2009
Received by editor(s) in revised form: December 17, 2009, and January 28, 2010
Published electronically: July 22, 2011
Additional Notes: The authors were supported by Dirección General de Investigación, Grants No. MTM2009-10418 and MTM2010-19821, and by “Programa de Ayudas a Grupos de Excelencia de la Región de Murcia”, Fundación Séneca, Agencia de Ciencia y Tecnología de la Región de Murcia (Plan Regional de Ciencia y Tecnología 2007/2010), 04540/GERM/06.
Article copyright: © Copyright 2011 American Mathematical Society

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