Slant Curves and Contact Magnetic Curves in Sasakian Lorentzian 3-Manifolds
Abstract
:1. Introduction
2. Preliminaries
Contact Lorentzian Manifold
3. Slant Curves in Contact Lorentzian Three-Manifolds
3.1. Lorentzian Cross Product
- (1)
- The Lorentzian cross product is bilinear and anti-symmetric.
- (2)
- is perpendicular both of X and Y.
- (3)
- .
- (4)
- (5)
- Define a mixed product by Then,
- (6)
3.2. Frenet Slant Curves
3.3. Null Slant Curves
4. Contact Magnetic Curves
- (i)
- a spacelike curve with spacelike normal vector field; or
- (ii)
- a timelike curve.
Example
Funding
Acknowledgments
Conflicts of Interest
References
- Cho, J.T.; Inoguchi, J.; Lee, J.-E. On slant curves in Sasakian 3-manifolds. Bull. Aust. Math. Soc. 2006, 74, 359–367. [Google Scholar] [CrossRef] [Green Version]
- Inoguchi, J.; Lee, J.-E. On slant curves in normal almost contact metric 3-manifolds. Beiträge Algebra Geom. 2014, 55, 603–620. [Google Scholar] [CrossRef]
- Baikoussis, C.; Blair, D.E. On Legendre curves in contact 3-manifolds. Geom. Dedic. 1994, 49, 135–142. [Google Scholar] [CrossRef]
- Barros, M.; Cabrerizo, J.L.; Fernandez, M.; Romero, A. The Gauss-Landau-Hall problem on Riemannian surfaces. J. Math. Phys. 2005, 46, 1–15. [Google Scholar] [CrossRef]
- Cabrerizo, J.L.; Fernandez, M.; Gomez, J.S. The contact Magnetic flow in 3D Sasakian manifolds. J. Phys. A Math. Theor. 2009, 42, 195201. [Google Scholar] [CrossRef]
- Calvaruso, G. Contact Lorentzian manifolds. Differ. Geom. Appl. 2011, 29, 541–551. [Google Scholar] [CrossRef]
- Calvaruso, G.; Perrone, D. Contact pseudo-metric manifolds. Differ. Geom. Appl. 2010, 28, 615–634. [Google Scholar] [CrossRef] [Green Version]
- Blair, D.E. Riemannian Geometry of Contact and Symplectic Manifolds. In Progress in Mathematics 203; Birkhäuser: Chicago, IL, USA, 2002. [Google Scholar]
- Ferrandez, A. Riemannian Versus Lorentzian submanifolds, some open problems. In Proceedings of the Workshop on Recent Topics in Differential Geometry, Santiago de Compostela, Spain, 16–19 July 1998; pp. 109–130. [Google Scholar]
- Inoguchi, J. Biharmonic curves in Minkowki 3-space. Int. J. Math. Math. Sci. 2003, 21, 1365–1368. [Google Scholar] [CrossRef]
- Camci, C. Extended cross product in a 3-dimensional almost contact metric manifold with applications to curve theory. Turk. J. Math. 2011, 35, 1–14. [Google Scholar]
- Duggal, K.L.; Jin, D.H. Null Curves and Hypersurfaces of Semi-Riemannian Manifolds; World Scientific Publishing: Singapore, 2007. [Google Scholar]
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Lee, J.-E. Slant Curves and Contact Magnetic Curves in Sasakian Lorentzian 3-Manifolds. Symmetry 2019, 11, 784. https://doi.org/10.3390/sym11060784
Lee J-E. Slant Curves and Contact Magnetic Curves in Sasakian Lorentzian 3-Manifolds. Symmetry. 2019; 11(6):784. https://doi.org/10.3390/sym11060784
Chicago/Turabian StyleLee, Ji-Eun. 2019. "Slant Curves and Contact Magnetic Curves in Sasakian Lorentzian 3-Manifolds" Symmetry 11, no. 6: 784. https://doi.org/10.3390/sym11060784