On Surfaces of Exceptional Lorentzian Lie Groups with a Four-Dimensional Isometry Group
Abstract
1. Introduction
- (I)
- Lorentzian Bianchi–Cartan–Vranceanu spaces, for which L is diagonal, show a structure similar to Riemannian BCV spaces (although the Lorentzian class presents more cases than its Riemannian analogue) and exhibit several interesting geometric properties. In particular, they are naturally reductive spaces and are defined by submersions over a pseudo-Riemannian surface of constant curvature [8].
- (II)
- An exceptional example , for which the minimal polynomial of L has a non-zero double root. At the Lie algebra level, this Lorentzian Lie group is described by
- (III)
- Some homogeneous plane waves, for which the minimal polynomial of L has 0 as a triple root. Homogeneous plane waves are a well-known topic, whose study which dates back to the work by [9] in the framework of theoretical physics. These examples are not naturally reductive.
2. Preliminaries
2.1. The Geometry of the Exceptional Example
2.2. Geometry of Surfaces in a Three-Dimensional Lorentzian Ambient Space
3. Codazzi Surfaces in
- (a)
- M is an integral surface of the distribution spanned by . This case only occurs if . M is parallel, flat and minimal, but not totally geodesic.
- (b)
- M is an integral surface of the distribution spanned by , where b and c are constants satisfying and . M is totally geodesic and has constant Gaussian curvature .
- (I)
- Every point of Σ admits an open neighborhood , where , for some smooth function . In particular, Σ is timelike.
- (II)
- Σ is flat.
- (III)
- Σ is CMC but never minimal.
- (IV)
- Σ is never parallel (in particular, totally geodesic).
4. Totally Umbilical Surfaces in
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Totally Umbilical | ✗ | ✓ | ✗ | ✗ |
Totally Geodesic | ✗ | ✓ | ✗ | ✓ |
Parallel | ✓ | ✓ | ✗ | ✓ |
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Calvaruso, G.; Pellegrino, L. On Surfaces of Exceptional Lorentzian Lie Groups with a Four-Dimensional Isometry Group. Mathematics 2025, 13, 2529. https://doi.org/10.3390/math13152529
Calvaruso G, Pellegrino L. On Surfaces of Exceptional Lorentzian Lie Groups with a Four-Dimensional Isometry Group. Mathematics. 2025; 13(15):2529. https://doi.org/10.3390/math13152529
Chicago/Turabian StyleCalvaruso, Giovanni, and Lorenzo Pellegrino. 2025. "On Surfaces of Exceptional Lorentzian Lie Groups with a Four-Dimensional Isometry Group" Mathematics 13, no. 15: 2529. https://doi.org/10.3390/math13152529
APA StyleCalvaruso, G., & Pellegrino, L. (2025). On Surfaces of Exceptional Lorentzian Lie Groups with a Four-Dimensional Isometry Group. Mathematics, 13(15), 2529. https://doi.org/10.3390/math13152529