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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Biharmonic lifts by means of pseudo-Riemannian submersions in dimension three


Authors: Miguel A. Javaloyes Victoria and Miguel A. Meroño Bayo
Journal: Trans. Amer. Math. Soc. 355 (2003), 169-176
MSC (2000): Primary 53C42; Secondary 53C50
Published electronically: September 11, 2002
MathSciNet review: 1928083
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the total lifts of curves by means of a submersion $\pi:M_s^3\rightarrow B_r^2$ that satisfy the condition $\Delta H=\lambda H$analyzing, in particular, the cases in which the submersion has totally geodesic fibres or integrable horizontal distribution. We also consider in detail the case $\lambda=0$ (biharmonic lifts). Moreover, we obtain a biharmonic lift in ${\mathbb R}^3$ by means of a Riemannian submersion that has non-constant mean curvature, getting so a counterexample to the Chen conjecture for ${\mathbb R}^3$ with a non-flat Riemannian metric.


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Additional Information

Miguel A. Javaloyes Victoria
Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain
Email: majava@um.es

Miguel A. Meroño Bayo
Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain
Email: mamb@um.es

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03119-7
PII: S 0002-9947(02)03119-7
Keywords: Pseudo-Riemannian submersions, biharmonic surfaces, Chen conjecture
Received by editor(s): March 12, 2002
Received by editor(s) in revised form: June 7, 2002
Published electronically: September 11, 2002
Additional Notes: This research has been partially supported by DGI Grant BFM2001-2871 (MCYT)
The first author was supported by a FPU Predoctoral Grant (MECD)
Article copyright: © Copyright 2002 American Mathematical Society