On Surfaces of Exceptional Lorentzian Lie Groups with a Four-Dimensional Isometry Group
In total, geodesic surfaces and their generalizations, namely totally umbilical and parallel surfaces, are well-known topics in Submanifold Theory and have been intensively studied in three-dimensional ambient spaces, both Riemannian and Lorentzian. In this paper, we prove the non-existence of paral...
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2025-08-01
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| author | Giovanni Calvaruso Lorenzo Pellegrino |
| author_facet | Giovanni Calvaruso Lorenzo Pellegrino |
| author_sort | Giovanni Calvaruso |
| collection | DOAJ |
| description | In total, geodesic surfaces and their generalizations, namely totally umbilical and parallel surfaces, are well-known topics in Submanifold Theory and have been intensively studied in three-dimensional ambient spaces, both Riemannian and Lorentzian. In this paper, we prove the non-existence of parallel and totally umbilical (in particular, totally geodesic) surfaces for three-dimensional Lorentzian Lie groups, which admit a four-dimensional isometry group, but are neither of Bianchi–Cartan–Vranceanu-type nor homogeneous plane waves. Consequently, the results of the present paper complete the investigation of these fundamental types of surfaces in all homogeneous Lorentzian manifolds, whose isometry group is four-dimensional. As a byproduct, we describe a large class of flat surfaces of constant mean curvature in these ambient spaces and exhibit a family of examples. |
| format | Article |
| id | doaj-art-9541afbbff284f5d98a8f43ccbcaad2d |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-9541afbbff284f5d98a8f43ccbcaad2d2025-08-20T03:04:43ZengMDPI AGMathematics2227-73902025-08-011315252910.3390/math13152529On Surfaces of Exceptional Lorentzian Lie Groups with a Four-Dimensional Isometry GroupGiovanni Calvaruso0Lorenzo Pellegrino1Dipartimento di Matematica e Fisica “E. De Giorgi”, Università del Salento, Prov. Lecce-Arnesano, 73100 Lecce, ItalyDipartimento di Matematica e Fisica “E. De Giorgi”, Università del Salento, Prov. Lecce-Arnesano, 73100 Lecce, ItalyIn total, geodesic surfaces and their generalizations, namely totally umbilical and parallel surfaces, are well-known topics in Submanifold Theory and have been intensively studied in three-dimensional ambient spaces, both Riemannian and Lorentzian. In this paper, we prove the non-existence of parallel and totally umbilical (in particular, totally geodesic) surfaces for three-dimensional Lorentzian Lie groups, which admit a four-dimensional isometry group, but are neither of Bianchi–Cartan–Vranceanu-type nor homogeneous plane waves. Consequently, the results of the present paper complete the investigation of these fundamental types of surfaces in all homogeneous Lorentzian manifolds, whose isometry group is four-dimensional. As a byproduct, we describe a large class of flat surfaces of constant mean curvature in these ambient spaces and exhibit a family of examples.https://www.mdpi.com/2227-7390/13/15/2529Lorentzian Lie groupsparallel surfacestotally umbilical surfacesCMC surfaces |
| spellingShingle | Giovanni Calvaruso Lorenzo Pellegrino On Surfaces of Exceptional Lorentzian Lie Groups with a Four-Dimensional Isometry Group Mathematics Lorentzian Lie groups parallel surfaces totally umbilical surfaces CMC surfaces |
| title | On Surfaces of Exceptional Lorentzian Lie Groups with a Four-Dimensional Isometry Group |
| title_full | On Surfaces of Exceptional Lorentzian Lie Groups with a Four-Dimensional Isometry Group |
| title_fullStr | On Surfaces of Exceptional Lorentzian Lie Groups with a Four-Dimensional Isometry Group |
| title_full_unstemmed | On Surfaces of Exceptional Lorentzian Lie Groups with a Four-Dimensional Isometry Group |
| title_short | On Surfaces of Exceptional Lorentzian Lie Groups with a Four-Dimensional Isometry Group |
| title_sort | on surfaces of exceptional lorentzian lie groups with a four dimensional isometry group |
| topic | Lorentzian Lie groups parallel surfaces totally umbilical surfaces CMC surfaces |
| url | https://www.mdpi.com/2227-7390/13/15/2529 |
| work_keys_str_mv | AT giovannicalvaruso onsurfacesofexceptionallorentzianliegroupswithafourdimensionalisometrygroup AT lorenzopellegrino onsurfacesofexceptionallorentzianliegroupswithafourdimensionalisometrygroup |