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Isometric immersions into a homogeneous Lorentzian Heisenberg group and rigidity

Published online by Cambridge University Press:  01 July 2009

SINUÊ DAYAN BARBERO LODOVICI  and
FERNANDO MANFIO
SINUÊ DAYAN BARBERO LODOVICI
Affiliation:
Departamento de Matemática Aplicada, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, CEP 05508-900, São Paulo, SP, Brazil. e-mail: [email protected] and [email protected]
FERNANDO MANFIO
Affiliation:
Departamento de Matemática Aplicada, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, CEP 05508-900, São Paulo, SP, Brazil. e-mail: [email protected] and [email protected]
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Abstract

In this paper we prove an existence result for local and global isometric immersions of semi-Riemannian surfaces into the three dimensional Heisenberg group endowed with a homogeneous left–invariant Lorentzian metric. As a corollary, we prove a rigidity result for such immersions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 1 , July 2009 , pp. 185 - 204
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

REFERENCES

[1]Dajczer, M.Submanifolds and Isometric Immersion. Mathematics Lecture Series. 13 (Publish or Perish, 1990).Google Scholar
[2]Daniel, B. Isometric immersions into S n × and H n × and applications to minimal surfaces. To appear in Trans. Amer. Math. Soc.Google Scholar
[3]Daniel, B.Isometric immersions into 3-dimensional homogeneous manifolds. Comment. Math. Helv. 82 (2007), 87–131.Google Scholar
[4]Dumitrescu, S. and Zeghib, A. Géométries Lorentziennes de dimension 3: classification et complétude. Preprint 2006 (arXiv:math/0703846).Google Scholar
[5]Falbel, E., Piccione, P. and Tausk, D. V. An isometric immersion theorem into contact sub-Riemannian spaceforms. Preprint 2008.Google Scholar
[6]Guediri, M.Sur la complétude des pseudo-métriques invariantes à gauche sur les groupes de Lie nilpotens. Rend. Sem. Mat. Univ. Politec., Torino. 52 (4) (1994), 371–376.Google Scholar
[7]Lodovici, S. D. B.An isometric immersion theorem in Sol3. Matemática Contemporânea. 30 (2006), 109–123.CrossRefGoogle Scholar
[8]Piccione, P. and Tausk, D. V.An existence theorem for G-structure preserving affine immersions. Indiana Univ. Math. J. 57 (2008), 1431–1465.Google Scholar
[9]Rahmani, N. and Rahmani, S.Lorentzian geometry of the Heisenberg group. Geom. Dedicata 118 (2006), 133–140.CrossRefGoogle Scholar
[10]Scott, P.The geometries of 3-manifolds. Bull. London Math. Soc. 15 (1983), 401–487.Google Scholar