A new representation of constant angle surfaces in H²×R with split quaternions

Abstract

In this paper we study surfaces in H²×R for which the unit normal makes a constant angle with the R-direction. The main idea is to show that constant angle surfaces in H²×R can be obtained by split quaternion product and the matrix representations. Also we give some related examples with their projections of figures.

Keywords

• Constant angle surfaces,product manifold,split quaternions,Minkowski space

References

1. Atalay, G.Ş., Güler, F. and Kasap, E., Spacelike Constant Angle Surfaces in Minkowski 3-space, J. Math. Comput. Sci. 2 (2012), No. 3, 451-461.
2. Aslan, S. and Yaylı, Y., Canal surfaces with quaternions. Adv. Appl. Clifford Algebras (2015).
3. Aslan, S. and Yaylı, Y., Split quaternions and canal surfaces in Minkowski 3-space. Int. J. Geom.5(2), (2016), 51--61.
4. Babaarslan, M. and Yaylı, Y., A new approach to constant slope surfaces with quaternion. ISRN Geom. 2012, Article ID 126358, p. 8 (2012). doi:10.5402/ 2012/126358
5. Babaarslan, M. and Yaylı, Y., Split quaternions and spacelike constant slope surfaces in Minkowski 3-space. Int. J. Geom. Vol. 2, No.1 (2013) 23--33.
6. Babaarslan, M. and Yaylı, Y., Split quaternions and timelike constant slope surfaces in Minkowski 3-space, arXiv:1206.3935 [math-ph]
7. Cermelli, P. and Di Scala A., J., Constant angle surfaces in liquid crystals. Phylos Magazine. 87, (2007) 1871-1888.
8. Dillen, F., Fastenakels, J., Van der Veken J. and Vrancken, L., Constant angle surfaces in S²×R. Monaths. Math. 152, (2007) 89--96.

9. Dillen, F. and Munteanu, M.,I., Constant Angle Surfaces in H²×R. Bull. Braz.Math. Soc. 40, (2009) 85--97.
10. Dillen, F. and Munteanu, M.I., Surfaces in H⁺×R, Pure and Applied Differential Geometry, PADGE 2007, Berichte aus der Mathematik (Shaker Verlag) Eds. Franki Dillen, Ignace Van de Woestyne, (2007) 185-193.
11. Gök, İ., Quaternionic approach of canal surfaces constructed by some new ideas, Adv. Appl. Clifford Algebras, 27 (2017) 1175--1190.
12. Gray, A., Abbena, E. and Salamon, S., Modern Geometry of Curves and Surfaces with Mathematica. 3rd Edition. Studies in Advanced Mathematics. 47, 2006.
13. Güler, F., Şaffak, G. and Kasap, E., Timelike Constant Angle Surfaces in Minkowski Space R₁³, Int. J. Contemp. Math. Sciences, Vol. 6, no. 44, (2011), 2189--2200.
14. Hacısalihoğlu, H.H., Hareket Geometrisi ve Kuaterniyonlar Teorisi. Faculty of Sciences and Arts, University of Gazi Press 1983.
15. Kocakusaklı, E., Tuncer, O.O, Gok, İ. and Yaylı Y., A New Representation of Canal Surfaces with Split Quaternions in Minkowski 3-Space Adv. Appl. Clifford Algebras, 27 (2017) 1387--1409.
16. Kula, L. and Yayli, Y., Split quaternions and rotations in semi Euclidean space E₂⁴. J. Korean Math. Soc. 44(6), (2007) 1313--1327.
17. Lopez, R., and Munteanu, M.I., Constant Angle Surfaces in Minkowski Space, Bulletin of the Belgian Mathematical Society, Volume18, Number 2 (2011), 271-286.
18. Munteanu, M.I. and Nistor, A. I., A New Approach on Constant Angle Surfaces in E³. Turk. J. Math. Vol. 33. (2009) 169-178.
19. Nistor, A., Certain Constant Angle Surfaces Constructed on Curves, Int. Electron. J. Geom., Vol. 4 No.1(2011), 79--87.
20. O'Neill, B., Semi-Riemannian Geometry. Academic Press. New York, 1983.
21. Özdemir, M. and Ergin, A.A., Rotations with unit timelike quaternions in Minkowski 3-space. J. Geom. Phys. 56(2), (2006) 322--326.
22. Özkaldı, S. and Yaylı, Y., Constant Angle Surfaces and Curves in E³, Int. Electron. J. Geom., Vol. 4, No.1 (2011), 70--78.
23. Ratcliffe, J. G., Foundatitons of Hyperbolic Manifolds, Springer-Verlag, New York, 1994.
24. Struik, D.J., Lectures on Classical Differential Geometry. New York. Dover, 1988.
25. Fastenakels, J., Munteanu M. I. and Van der Veken, J., Constant angle surfaces in the Heisenberg group, Acta Math. Sinica - Engl. Series, 27, 4 (2011), 747--756.
26. Onnis I. and Piu P., Constant angle surfaces in the Lorentzian Heisenberg group, Arch. Math. 109 (2017), 575--589.
27. Jia, Yan-Bin, Quaternion and Rotation, Com S 477/577 Notes Sep 5, 2017.
28. Ward, J.P., Quaternions and Cayley Numbers. Kluwer Academic Publishers, Boston 1997.