Singularities of the Darboux ruled surface of a space curve in the pseudo-Galilean space

Öz

In this article, we establish the singularity theory in a pseudo-Galilean space , a special case of Cayley-Klein spaces. We consider the cases where the Darboux ruled surface in  is diffeomorphic to some surfaces in the neighbourhood of a singular point.  In addition, we investigate the relationship between singularities of discriminant, bifurcation sets of the function, and geometric invariants of curves in . 

Anahtar Kelimeler

• Height function,singularities,Darboux ruled surface

Kaynakça

1. [1] Bruce, J.W.; Giblin, P.J. Curves and Singularities, second edition, Cambridge Univ.Press, 1992.
2. [2] Giering, O. Vorlesungen Über Höhere Geometrie, Fri-edr. Vieweg & Sohn, Braunschweig, Germany, 1982. [3] Arnol'd, V.I.; Gusein-Zade, S.M.; Varchenko, A.N. Singu-larities of Differentiable Maps vol. I, Birkhäuser, 1986.
3. [4] Mond, D. Singularities of the tangent developable surfa-ce of aspace curve, Quart. J. Math. Oxford, 1989; 40, 79-91.
4. [5] Sahin, T.; Yılmaz, M. On singularities of the Galilean spherical darboux ruled surface of a space curve in , Ukranian Math. Journal, 2011; 62(10), 1597-1610. [6] Sahin, T.; Yılmaz, M. The rectifying developable and the tangent indicatrix of a curve in Galilean 3-space, Acta Math. Hungar. 2011; 132(1-2), 154-167.
5. [7] Izumiya, S.; Katsumi, H.; Yamasaki, T. The Rectifying developable and the spherical darboux image of a space curve, Geometry and topology of caustics '98 -- Banach Center Publications. 1999; 50, 137-149.
6. [8] Che, M.G.; Jiang, Y.; Pei, D. The hyperbolic Darboux image and rektifying Gaussian surface of nonlightlike curve in Minkowski 3-space, Journal of Math. Research & Exposition. 2008; 28, 651-658.
7. [9] Molnar, E. The projective interpretation of the eight 3-dimensional homogeneous geometries, Beitr. Algebra Geom. 1997; 38, 261-288.
8. [10] Erjavec, Z.; Divjak, B. The general solution of the Fre-net system of differential equations for curves in the pseu-do-Galilean space , Mathematical Communications. 1997; 2, 143-147. [11] Divjak, B. Geometrija psudogalilejevih prostora, Ph.D. Thesis, University of Zagreb. 1997.

9. [12] Divjak, B.; Šipuš, Ž.M. Special curves on ruled surfaces in Galilean and pseudo-Galilean spaces, Acta Math. Hun-gar., 2003; 98(3), 203-215.
10. [13] Yaglom, I.M. A Simple Non-Euclidean Geometry and Physical Basis, Springer-Verlag Newyork. 1979.