The space of Lorentzian flat tori in anti-de Sitter $3$-space
HTML articles powered by AMS MathViewer
- by María A. León-Guzmán, Pablo Mira and José A. Pastor PDF
- Trans. Amer. Math. Soc. 363 (2011), 6549-6573 Request permission
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.References
- Juan A. Aledo, José A. Gálvez, and Pablo Mira, Isometric immersions of $\Bbb L^2$ into $\Bbb L^4$, Differential Geom. Appl. 24 (2006), no. 6, 613–627. MR 2280054, DOI 10.1016/j.difgeo.2006.04.006
- Juan A. Aledo, José A. Gálvez, and Pablo Mira, A D’Alembert formula for flat surfaces in the 3-sphere, J. Geom. Anal. 19 (2009), no. 2, 211–232. MR 2481959, DOI 10.1007/s12220-008-9057-4
- Manuel Barros, Angel Ferrández, Pascual Lucas, and Miguel A. Meroño, Hopf cylinders, $B$-scrolls and solitons of the Betchov-Da Rios equation in the three-dimensional anti-de Sitter space, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 4, 505–509 (English, with English and French summaries). MR 1351107
- M. Barros, A. Ferrández, P. Lucas, M.A. Meroño, Solutions of the Betchov-Da Rios soliton equation in the anti-de Sitter $3$-space. New Approaches in Nonlinear Analysis. Hadronic Press Inc., Palm Harbor, Florida, 1999, pp. 51–71.
- L. Bianchi, Sulle superficie a curvatura nulla in geometria ellittica, Ann. Mat. Pura Appl., 24 (1896), 93–129.
- Thomas E. Cecil, On the completeness of flat surfaces in $S^{3}$, Colloq. Math. 33 (1975), no. 1, 139–143. MR 467611, DOI 10.4064/cm-33-1-139-143
- Marcos Dajczer and Katsumi Nomizu, On flat surfaces in $S^{3}_{1}$ and $H^{3}_{1}$, Manifolds and Lie groups (Notre Dame, Ind., 1980) Progr. Math., vol. 14, Birkhäuser, Boston, Mass., 1981, pp. 71–108. MR 642853
- Jiri Dadok and Ji-Ping Sha, On embedded flat surfaces in $S^3$, J. Geom. Anal. 7 (1997), no. 1, 47–55. MR 1630773, DOI 10.1007/BF02921704
- José A. Gálvez and Pablo Mira, Isometric immersions of $\Bbb R^2$ into $\Bbb R^4$ and perturbation of Hopf tori, Math. Z. 266 (2010), no. 1, 207–227. MR 2670680, DOI 10.1007/s00209-009-0564-1
- Masatoshi Kokubu, Masaaki Umehara, and Kotaro Yamada, Flat fronts in hyperbolic 3-space, Pacific J. Math. 216 (2004), no. 1, 149–175. MR 2094586, DOI 10.2140/pjm.2004.216.149
- Masatoshi Kokubu, Wayne Rossman, Kentaro Saji, Masaaki Umehara, and Kotaro Yamada, Singularities of flat fronts in hyperbolic space, Pacific J. Math. 221 (2005), no. 2, 303–351. MR 2196639, DOI 10.2140/pjm.2005.221.303
- Yoshihisa Kitagawa, Periodicity of the asymptotic curves on flat tori in $S^3$, J. Math. Soc. Japan 40 (1988), no. 3, 457–476. MR 945347, DOI 10.2969/jmsj/04030457
- Yoshihisa Kitagawa, Embedded flat tori in the unit $3$-sphere, J. Math. Soc. Japan 47 (1995), no. 2, 275–296. MR 1317283, DOI 10.2969/jmsj/04720275
- Ettore Minguzzi and Miguel Sánchez, The causal hierarchy of spacetimes, Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., Eur. Math. Soc., Zürich, 2008, pp. 299–358. MR 2436235, DOI 10.4171/051-1/9
- Satoko Murata and Masaaki Umehara, Flat surfaces with singularities in Euclidean 3-space, J. Differential Geom. 82 (2009), no. 2, 279–316. MR 2520794
- Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
- Shigeo Sasaki, On complete surfaces with Gaussian curvature zero in $3$-sphere, Colloq. Math. 26 (1972), 165–174. MR 348677, DOI 10.4064/cm-26-1-165-174
- Michael Spivak, A comprehensive introduction to differential geometry. Vol. IV, Publish or Perish, Inc., Boston, Mass., 1975. MR 0394452
- Kentaro Saji, Masaaki Umehara, and Kotaro Yamada, The geometry of fronts, Ann. of Math. (2) 169 (2009), no. 2, 491–529. MR 2480610, DOI 10.4007/annals.2009.169.491
- Joel L. Weiner, Flat tori in $S^3$ and their Gauss maps, Proc. London Math. Soc. (3) 62 (1991), no. 1, 54–76. MR 1078213, DOI 10.1112/plms/s3-62.1.54
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
- MR Author ID: 692410
- Email: pablo.mira@upct.es
- José A. Pastor
- Affiliation: Departamento de Matemáticas, Universidad de Murcia, Spain
- Email: josepastor@um.es
- 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.
- © Copyright 2011 American Mathematical Society
- 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
- MathSciNet review: 2833568