Constant curvature surfaces in a pseudo-isotropic space

Article Sidebar

PDF
Published: Sep 30, 2018
DOI: https://doi.org/10.5556/j.tkjm.49.2018.2613
Keywords:
Pseudo-isotropic space surface of revolution Gaussian curvature mean curvature

Main Article Content

Muhittin Evren Aydin

Abstract

In this study, we deal with the local structure of curves and surfaces immersed in a pseudo-isotropic space $\mathbb{I}_{p}^{3}$ that is a particular Cayley-Klein space. We provide the formulas of curvature, torsion and Frenet trihedron for spacelike and timelike curves, respectively. The causal character of all admissible surfaces in $\mathbb{I}_{p}^{3}$ has to be timelike up to its absolute. We introduce the formulas of Gaussian and mean curvature for timelike surfaces in $\mathbb{I}_{p}^{3}$. As applications, we describe the surfaces of revolution which are the orbits of a plane curve under a hyperbolic rotation with constant Gaussian and mean curvature.

Article Details

How to Cite
Aydin, M. E. (2018). Constant curvature surfaces in a pseudo-isotropic space. Tamkang Journal of Mathematics, 49(3), 221–233. https://doi.org/10.5556/j.tkjm.49.2018.2613
Issue
Vol. 49 No. 3 (2018)
Section
Papers
Author Biography

Muhittin Evren Aydin

Department of Mathematics, Faculty of Science, Firat University, Elazig, 23119, Turkey.

References

M. E. Aydin and A. Mihai, Ruled surfaces generated byelliptic cylindrical curves in the isotropic space, Georgian Math. J. 2017;doi.org/10.1515/gmj-2017-0044.

M. E. Aydin and I. Mihai, On certain surfaces in the isotropic 4-space, Math. Commun. 22(1) (2017), 41--51.

M. E. Aydin, Classification results on surfaces in the isotropic 3-space, AKU J. Sci. Eng., 16(2016), 239--246.

B. Y. Chen, S. Decu and L. Verstraelen, Notes on isotropic geometry of production models, Kragujevac J. Math., 38(1) (2014),23--33.

B. Y. Chen, Mean curvature and shape operator of isometric immersions in real-space-forms, Glasg. Math. J., 38(1996), 87--97.

B. Y. Chen, Pseudo-Riemannian geometry, $delta $-Invariants and applications, World Scientific, Singapore, 2011.

M. Crasmareanu, Cylindrical Tzitzeica curves implies forced harmonic oscillators, Balkan J. Geom. Appl., 7(1)(2002),37--42.

L. C. B. Da Silva, Rotation minimizing frames and spherical curves in simply isotropic and semi-isotropic 3-spaces,arXiv:1707.06321v2 [math.DG].

L. C. B. Da Silva, The geometry of Gauss map and shape operator in simply isotropic and pseudo-isotropic spaces, arXiv:1801.01187v1 [math.DG].

B. Divjak, Geometrija pseudogalilejevih prostora (Ph.D. thesis), University of Zagreb, 1997.

B. Divjak, Curves in pseudo-Galilean geometry, Annales Universitatis Scientiarum Budapestinensis de Rolando Eotvos Nominatae,41(1998), 117--128.

Z. Erjavec, B. Divjak and D. Horvat, The general solutions of Frenet's system in the equiform geometry of the Galilean, pseudo-Galilean, simple isotropic and double isotropic space, Int. Math. Forum 6(17) (2011), 837-856.

O. Giering, Vorlesungenuber hohere Geometrie, Friedr. Vieweg & Sohn, Braunschweig, Germany, 1982.

M. Husty and O. Roschel, On a particular class of cyclides in isotropic respectively pseudoisotropic space, Coll. Math. Soc. J. Bolyai, 46(1984), 531--557.

M. K. Karacan, D. W. Yoon and S. Kiziltug, Helicoidal surfaces in the three dimensional simply isotropic space $I_{1}^{3}$, Tamkang J. Math., 48(2017), 123--134.

D. Klawitter, Clifford Algebras: Geometric Modelling and Chain Geometries with Application in Kinematics, Springer Spektrum, 2015.

R. Lopez, Differential Geometry of curves and surfaces

in Lorentz-Minkowski space, Int. Electron. J. Geom., 7(2014), 44--107.

F. Meszaros, Die Zykliden 3. Ordnung im pseudoisotropen Raum II, Math. Pannonica 4(2)(1993), 273--285.

F. Meszaros, Klassifikationstheorie der verallgemeinerten zykliden 4. ordnung in pseudoisotropen Raum, Math. Pannonica, 18(2)(2007), 299--323.

A. Mihai, Geometric inequalities for purely real submanifolds in complex space forms}, Results Math., 55(2009),457--468.

I. Mihai, On the generalized Wintgen inequality for Lagrangian submanifolds in complex space forms, Nonlinear Analysis 95(2014), 714-720.

Z. Milin-Sipus and B. Divjak, Surfaces of constant curvature in the pseudo-Galilean space, Int. J. Math. Sci., 2012, Art ID375264, 28pp.

Z. Milin-Sipus, Translation surfaces of constant curvatures in a simply isotropic space, Period. Math. Hung., 68(2014), 160--175.

A. O. Ogrenmis, Rotational surfaces in isotropic spaces satisfying Weingarten conditions, Open Physics 14(9) (2016), 221--225.

O. Ogrenmis, M. Bektas and M. Ergut, On helices in the doubly isotropic space $I_{3}^{left( 2right) }$, Int. Math. Forum 1(13) (2006),623--627.

B. O`Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.

A. Onishchick, R. Sulanke, Projective and Cayley-Klein Geometries, Springer, 2006.

H. Sachs, Isotrope Geometrie des Raumes, Vieweg Verlag, Braunschweig, 1990.

K. Strubecker, Differentialgeometrie des isotropen Raumes III}, Flachentheorie, Math. Zeitsch., 48(1942), 369--427.

I. M. Yaglom, A simple non-Euclidean Geometry and Its Physical Basis, An elementary account of Galilean geometry and the Galilean principle of relativity, Heidelberg Science Library. Translated from the Russian by

Abe Shenitzer. With the editorial assistance of Basil Gordon.

Springer-Verlag, New York-Heidelberg, 1979.

D.W. Yoon, J.W. Lee, Linear Weingarten helicoidal

surfaces in isotropic space, Symmetry 2016, 8(11), 126;

doi:10.3390/sym8110126.