Parallel Surfaces of Spacelike Ruled Weingarten Surfaces in Minkowski 3-space

Abstract

In this work, it is shown that parallel surfaces of spacelike ruled surfaces which are developable are spacelike ruled Weingarten surfaces. It is also shown that parallel surfaces of non-developable ruled Weingarten surfaces are again Weingarten surfaces. Finally, some properties of that kind parallel surfaces are obtained in Minkowski 3-space

Keywords

• Spacelike Weingarten surface,parallel surface,spacelike ruled Weingarten surface,curvatures

Parallel Surfaces of Spacelike Ruled Weingarten Surfaces in

Öz

References

1. Beltrami, E., Risoluzione di un problema relativo alla teoria delle supercie gobbe, Ann. Mat. Pura Appl., 7, 139-150., 1865.
2. Brunt, B., Weingarten surfaces design and application of curves and surfaces, Fisher, R., (Ed.), Mathematics of surfaces V, Oxford Univ. Press., 1994.
3. Brunt, B. and Grant, K., Potential applications of Weingarten surfaces in CAGD. I: Weingarten surfaces and surface shape investigation, Comput. Aided Geom. Des., 13, 569-582., 1996.
4. Dillen, F. and Kühnel, W., Ruled Weingarten surfaces in Minkowski 3-space, Manuscripta Math., 98, 307-320., 1999.
5. Dillen, F. and Sodsiri, W., Ruled surfaces of Weingarten type in Minkowski 3-space, J. Geom., 83, 10-21., 2005.
6. Dillen, F. and Sodsiri, W., Ruled surfaces of Weingarten type in Minkowski 3-space-II, J. Geom., 84, 37-44., 2005.
7. Dini, U., Sulle super.cie gobbe nelle quali uno dei due raggidi curvatura principale e una funzione dellaltro, Ann. Mat. Pura Appl., 7, 205-210.,1865.
8. Görgülü, A. and Çöken, C., The Dupin indicatrix for parallel pseudo-Euclidean hypersurfaces in pseudo-Euclidean space in

9. Journ. Inst. Math. and Comp. Sci. (Math Series), 7(3), 221-225., 1994.
10. semi-Euclidean space
11. Gray, A., Modern differential geometry of curves and surfaces, CRC Press, Inc., 1993.
12. Hou, Z. H. and Ji, F., Helicoidal surfaces with
13. Kim, M. H. and Yoon, D. W., Weingarten quadric surfaces in a Euclidean 3-space, Turk. J. Math. 34, 1-7., 2010.
14. Koch, R., Die Weingarten-regelflächen, J. Geom. 47, 77-85., 1993.
15. Kühnel, W., Ruled W-surfaces, Arch. Math. (Basel), 62, 475-480., 1994.
16. Lopez, R., Di¤erential geometry of curves and surfaces in Lorentz-Minkowski space, Mini-Course taught at the Instituto de Matematica e Estatistica (IME-USP), University of Sao Paulo, Brasil, 2008.
17. O’Neill, B., Semi Riemannian geometry with applications to relativity, Academic Press, Inc. New York, 1983.
18. Turgut, A. and Hacısalihoğlu, H. H., Spacelike ruled surfaces in the Minkowski 3-space, Commun. Fac. Sci. Univ. Ank. Series A1, v. 46, pp. 83-91, 1997.
19. Unluturk, Y., and Ekici, C., On spacelike parallel ruled surfaces in Minkowski 3-space, (Preprint).
20. Unluturk, Y., Ekici, C., and Özüsağlam, E., On parallel surfaces in Minkowski 3-space, (Preprint).
21. Weingarten, J., Über eine klasse auf einander abwickelbarer .ächen, J. Reine Angew. Math. 59, 382-393, 1861.
22. Weingarten, J., Über eine flächen, derer normalen eine gegebene flächeberühren, Journal für die Reine und Angewandte Mathematik, 62, 61-63, 1863.
23. D. W. Yoon, Some properties of parallel surfaces in Euclidean 3-spaces, Honam Mathematical J. 30, No. 4, pp, 637-644, 2008.