Browse > Article
http://dx.doi.org/10.5831/HMJ.2018.40.4.701

CMC SURFACES FOLIATED BY ELLIPSES IN EUCLIDEAN SPACE E3  

Ali, Ahmad Tawfik (Department of Mathematics, Faculty of Science, King Abdul Aziz University)
Publication Information
Honam Mathematical Journal / v.40, no.4, 2018 , pp. 701-718 More about this Journal
Abstract
In this paper, we will study the constant mean curvature (CMC) surfaces foliated by ellipses in three dimensional Euclidean space $E^3$. We prove that: (1): Surfaces foliated by ellipses are CMC surfaces if and only if it is a part of generalized cylinder. (2): All surfaces foliated by ellipses are not minimal surfaces. (3): CMC surfaces foliated by ellipses are developable surfaces. (4): CMC surfaces foliated by ellipses are translation surfaces generated by a straight line and plane curve.
Keywords
Cyclic surfaces; Euclidean space; Gaussian and Mean Curvatures;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 A. T. Ali, H. S. Abdel Aziz and A. H. Sorour, On curvatures and points of the translation surfaces in Euclidean 3-space, J. Egyptian Math. Soc., 23 (2015), 167-172 .   DOI
2 A. T. Ali and F. M. Hamdoon, Surfaces foliated by ellipses with constant Gaussian curvature in Euclidean 3-space, Korean J. Math., 25(4) (2017), 537-554.   DOI
3 M. Barros, General helices and a theorem of Lancret, Proc. Amer. Math. Soc., 125 (1997), 1503-1509.   DOI
4 M. Cetin, Y. Tuncer and N. Ekmekci, Translation surfaces in Euclidean 3-space, World Academy Sci. Engin. Tech., 52 (2011), 864-868.
5 C. Delaunay, Sur la surface de revolution dont la courbure moyenne est constante, J. Math. Pure Appl., 6 (1841), 309-320.
6 A. Enneper, Ueber die cyclischen Flachen, Nach. Konigl. Ges. d. Wisseensch. Gottingen Math.Phys., Kl (1866), 243-249.
7 R. Lopez, Cyclic surfaces of constant Gauss curvature, Houston J. Math., 27(4) (2001), 799-805.
8 R. Lopez, On linear Weingarten surfaces, Int. J. Math., 19 (2008), 439-448.   DOI
9 R. Lopez, Special Weingarten surfaces foliated by circles, Monatsh. Math., 154 (2008), 289-302.   DOI
10 I. M. Munteanu and A. I. Nistor, On the geometry of the second fundamental form of translation surfaces in $E^3$, Houston J. Math., 37 (2011), 1087-1102.
11 J. C. Nitsche, Cyclic surfaces of constant mean curvature, Nachr. Akad. Wiss. Gottingen Math. Phys., II 1 (1989), 1-5.
12 A. T. Ali, Position vectors of slant helices in Euclidean 3-space, J. Egyptian Math. Soc. 20(1) (2012), 1-6.   DOI
13 B. Riemann, "Uber die Flachen vom kleinsten Inhalt bei gegebener Begrenzung, Abh. Konigl. Ges. d. Wisseensch. Gottingen Mathematica, Cl:13 (1868), 329-333.
14 D. J. Struik, Lectures in classical Differential Geometry, Addison, -Wesley, Reading, MA, 1961.
15 A. Enneper, Die cyclischen Flachen, Z. Math. Phys., 14 (1869), 393-421.
16 A. T. Ali, Position vectors of general helices in Euclidean 3-space, Bull. Math. Anal. Appl., 3(2) (2010), 198-205.