Abstract
In this work, we give a characterization of pseudo-parallel Lorentzian surfaces with non-flat normal bundle in pseudo-Riemannian space forms as \(\uplambda \)-isotropic surfaces, extending an analogous result by Asperti–Lobos–Mercuri for the pseudo-parallel case in Riemannian space forms. In addition, for this kind of pseudo-parallel surfaces we give a characterization using the concept of hyperbola of curvature. In particular, we get a non-existence result for pseudo-parallel Lorentzian surfaces with non-flat normal bundle in Lorentzian space forms. Moreover, in codimension two, we show that locally any pseudo-parallel Lorentzian surface with non-flat normal bundle and constant pseudo-parallelism function is congruent to a piece of a Lorentzian surface of the Veronese type. Finally, an example of an extremal and flat pseudo-parallel Lorentzian surface with non-flat normal bundle which is not a semi-parallel surface is given in codimension three.
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Acknowledgements
The authors are thankful to the referees for their valuable comments and suggestions towards the improvement of this work. The authors are also thankful to R. Tojeiro and M. Tassi for useful discussions.
Funding
Guillermo Lobos: partially supported by FAPESP Proc. No. 16/23746-6. Mynor Melara: partially supported by CNPq Proc. No. 141496/2020-7. Oscar Palmas: partially supported by UNAM under Project PAPIIT-DGAPA IN101322.
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Lobos, G., Melara, M. & Palmas, O. Pseudo-parallel Lorentzian Surfaces in Pseudo-Riemannian Space Forms. Results Math 78, 39 (2023). https://doi.org/10.1007/s00025-022-01808-z
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DOI: https://doi.org/10.1007/s00025-022-01808-z
Keywords
- Pseudo-parallel surface
- \(\uplambda \)-isotropic surface
- extremal Lorentzian surface
- hyperbola of curvature