Ruled Surfaces Created with Spherical Curves in Pseudo-Galilean Space
Abstract
This study focuses on ruled surfaces in three-dimensional pseudo-Galilean space G_3^1, where the generator vector field of these ruled surfaces is defined by a curve on the unit pseudo-Galilean sphere. Initially, the orthonormal frame and derivative equations of the curve on the pseudo-Galilean unit sphere are calculated. Conditions are provided for a unit speed curve on a unit pseudo-Galilean sphere to be curvature line, geodesic curve or asymptotic curve in the pseudo-Galilean sense. Subsequently, the structure functions and ruled invariants of the ruled surfaces are defined and obtained in the pseudo-Galilean sense. This research encompasses all three types of ruled surfaces, and the relationships between the Frenet frames of the generating curves and those of the corresponding surfaces are systematically examined.
Keywords
Pseudo-Galilean space, pseudo-galilean sphere, ruled surface, spherical curve
Supporting Institution
No supporting institution
Ethical Statement
No approval from the Board of Ethics is required.
References
- M. J. Greenberg, Euclidean and non-Euclidean geometries: Development and history, 4th Edition, W. H. Freeman, 2008.
- R. Bonola, Non-Euclidean Geometry: A Critical and Historical Study of Its Development, Dover Books on Mathematics Series, Dover Publications, 1955.
- B. O’Neill, Semi-Riemannian geometry with applications to relativity, Pure and Applied Mathematics, Academic Press, 1983.
- A. Einstein, The foundation of the general theory of relativity, Annalen der Physik 354 (7) (1916) 769–822.
- I. M. Yaglom, An elementary account of Galilean geometry and the Galilean principle of relativity, A Simple Non-Euclidean Geometry and Physical Basis, Heidelberg Science Library, 1st Edition, Springer, 1979.
- O. Röschel, Die Geometrie des Galileischen Raumes, Habilitationsschrift, Montanuniversität Leoben, 1984.
- B. Divjak, Curves in pseudo-Galilean geometry, Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae, Sectio Mathematica 41 (1998) 117–128.
- B. Divjak, Z. M. Šipuš, Special curves on the ruled surfaces in Galilean and pseudo-Galilean spaces, Acta Mathematica Hungarica 98 (2002) 203–215.
- B. Divjak, Z. M. Šipuš, Some special surfaces in the pseudo-Galilean space, Acta Mathematica Hungarica, 118 (3) (2008) 209–226.
- M. P. Do Carmo, Differential geometry of curves and surfaces, 1st Edition, Pearson, 1976.