Hostname: page-component-cb9f654ff-mx8w7 Total loading time: 0 Render date: 2025-09-02T09:18:47.228Z Has data issue: false hasContentIssue false
  • English
  • Français

LORENTZIAN BCV SPACES: PROPERTIES AND GEOMETRY OF THEIR SURFACES

Part of: Global differential geometry Local differential geometry

Published online by Cambridge University Press:  25 July 2025

GIOVANNI CALVARUSO Open the ORCID record for GIOVANNI CALVARUSO [Opens in a new window]  and
LORENZO PELLEGRINO Open the ORCID record for LORENZO PELLEGRINO [Opens in a new window]
GIOVANNI CALVARUSO*
Affiliation:
Dipartimento di Matematica e Fisica ‘E. De Giorgi’ https://ror.org/03fc1k060Università del Salento, Prov. Lecce-Arnesano, 73100 Lecce, Italy
LORENZO PELLEGRINO
Affiliation:
Dipartimento di Matematica e Fisica ‘E. De Giorgi’ https://ror.org/03fc1k060Università del Salento, Prov. Lecce-Arnesano, 73100 Lecce, Italy e-mail: lorenzo.pellegrino@unisalento.it
*
e-mail: giovanni.calvaruso@unisalento.it
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the Lorentzian analogues of Riemannian Bianchi–Cartan–Vranceanu spaces. We provide their general description and emphasize their role in the classification of three-dimensional homogeneous Lorentzian manifolds with a four-dimensional isometry group. We then illustrate their geometric properties (with particular regard to curvature, Killing vector fields and their description as Lorentzian Lie groups) and we study several relevant classes of surfaces (parallel, totally umbilical, minimal, constant mean curvature) in these homogeneous Lorentzian three-manifolds.

Keywords

Lorentzian manifoldsBianchi–Cartan–Vranceanu spaceshomogeneous spacestotally umbilical surfacesparallel surfacesminimal and CMC surfaces

MSC classification

Primary: 53B25: Local submanifolds
Secondary: 53C50: Lorentz manifolds, manifolds with indefinite metrics

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , First View , pp. 1 - 23
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Communicated by Graeme Wilkin

References

Belkhelfa, M., Dillen, F. and Inoguchi, J., ‘Surfaces with parallel second fundamental form in Bianchi–Cartan–Vranceanu spaces’, Banach Center Publ. 57 (2002), 67–87.10.4064/bc57-0-5CrossRefGoogle Scholar
Bianchi, L., Lezioni di Geometrie Differenziale (E. Spoerri Librao-Editore, Pisa, 1894).Google Scholar
Bianchi, L., Lezioni sulla teoria dei gruppi continui e finiti di trasformazioni (Ed. Zanichelli, Bologna, 1928).Google Scholar
Calvaruso, G., Castrillón-Lopez, M. and Pellegrino, L., ‘On totally umbilical and minimal surfaces of the Lorentzian Heisenberg groups’, Math. Nachr. 298 (2025), 1922–1942.10.1002/mana.12020CrossRefGoogle Scholar
Calvaruso, G., Onnis, I. I., Pellegrino, L. and Uccheddu, D., ‘Helix surfaces for Berger-like metrics on the anti-de Sitter space’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 118 (2024), 54, 29 pages.10.1007/s13398-024-01550-xCrossRefGoogle Scholar
Calvaruso, G. and Pellegrino, L., ‘Totally umbilical surfaces of Lorentzian reducible spaces’, J. Geom. Anal. 35 (2025) 131, 22 pages.10.1007/s12220-025-01959-9CrossRefGoogle Scholar
Calvaruso, G. and Perrone, D., ‘Metrics of Kaluza–Klein type on the anti-de Sitter space ${\mathbb{H}}_1^3$ ’, Math. Nachr. 287 (2014), 885–902.10.1002/mana.201200105CrossRefGoogle Scholar
Calvaruso, G. and Van der Veken, J., ‘Parallel surfaces in three-dimensional Lorentzian lie groups’, Taiwanese J. Math. 14 (2010), 223–250.10.11650/twjm/1500405737CrossRefGoogle Scholar
Calvaruso, G. and Zaeim, A., ‘Three-dimensional homogeneous Lorentzian structures’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 119 (2025), 41, 24 pages.10.1007/s13398-025-01707-2CrossRefGoogle Scholar
Calviño-Louzao, E., Ferreiro-Subrido, M., Garcia-Rio, E. and Vazquez-Lorenzo, R., ‘Homogeneous Riemannian structures in dimension three’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 70 (2023), 117, 11 pages.Google Scholar
Cartan, E., Leçons sur la Géométrie des Espaces de Riemann (Gauthier-Villars, Paris, 1928).Google Scholar
Daniel, B., ‘Isometric immersions into 3-dimensional homogeneous manifolds’, Comment. Math. Helv. 82 (2007), 87–131.10.4171/cmh/86CrossRefGoogle Scholar
Onnis, I. I., Passamani, A. P. and Piu, P., ‘Constant angle surfaces in Lorentzian Berger spheres’, J. Geom. Anal. 29 (2019), 1456–1478.10.1007/s12220-018-0044-0CrossRefGoogle Scholar
Onnis, I. I. and Piu, P., ‘Constant angle surfaces in the Lorentzian Heisenberg group’, Arch. Math. (Basel) 109 (2017), 575–589.10.1007/s00013-017-1104-6CrossRefGoogle Scholar
Rahmani, N. and Rahmani, S., ‘Structures homogenes Lorentziennes Sur le groupe de Heisenberg. I’, J. Geom. Phys. 13 (1994), 254–258.10.1016/0393-0440(94)90033-7CrossRefGoogle Scholar
Rahmani, S., ‘Métriques de Lorentz Sur les groupes de lie unimodulaires de dimension trois’, J. Geom. Phys. 9 (1992), 295–302.10.1016/0393-0440(92)90033-WCrossRefGoogle Scholar
Souam, R. and Toubiana, E., ‘Totally umbilic surfaces in homogeneous $3$ -manifolds’, Comment. Math. Helv. 84 (2009), 673–704.10.4171/cmh/177CrossRefGoogle Scholar
Turhan, E. and Altay, G., ‘Minimal surfaces in three dimensional Lorentzian Heisenberg group’, Beiträge Alg. Geom. 55 (2014), 1–23 10.1007/s13366-013-0156-4CrossRefGoogle Scholar
Vranceanu, G., Leçons de Géométrie Différentielle I (Ed. Acad. Rep. Roum, Bucarest, 1947).Google Scholar
Yildirim, A., ‘On Lorentzian BCV spaces’, Int. J. Math. Archive 3 (2012), 1365–1371.Google Scholar