Korean J. Math.  Vol 25, No 4 (2017)  pp.537-554
DOI: http://dx.doi.org/10.11568/kjm.2017.25.4.537

Surfaces foliated by ellipses with constant Gaussian curvature in Euclidean 3-space

Ahmed T. Ali, Fathi Mohamed Hamdoon

Abstract


In this paper, we study the surfaces foliated by  ellipses in three dimensional Euclidean space $\mathbf{E}^3$. We prove the following results: \textbf{(1)} The surface  foliated by an ellipse have constant Gaussian curvature $K$ if and only if the surface is flat, i.e. $K=0$. \textbf{(2)} The surface foliated by an ellipse is a flat if and only if it is   a part of generalized cylinder or part of generalized cone.


Keywords


Surfaces in Euclidean space, Gaussian curvature.

Subject classification

53A05, 53A17.

Sponsor(s)



Full Text:

PDF

References


Ali A.T., Position vectors of general helices in Euclidean 3-space, Bull. Math. Anal. Appl. 3 (2) (2010), 198–205. (Google Scholar)

Ali A.T., Position vectors of slant helices in Euclidean 3-space, J. Egyptian Math. Soc. 20 (1) (2012), 1–6. (Google Scholar)

Delaunay C., Sur la surface de r ́evolution dont la courbure moyenne est constante, J. Math. Pure Appl. 6 (1841), 309–320. (Google Scholar)

Enneper A., Ueber die cyclischen Fl ̈achen, Nach. K ̋onigl. Ges. d. Wisseensch. G ̋ottingen, Math. Phys. Kl (1866), 243–249. (Google Scholar)

Enneper A., Die cyclischen Fl ̈achen, Z. Math. Phys. 14 (1869), 393–421. (Google Scholar)

Lo ́pez R. Cyclic surfaces of constant Gauss curvature, Houston J. Math. 27 (4) (2001), 799–805. (Google Scholar)

L ́opez R. On linear Weingarten surfaces, Int. J. Math. 19 (2008), 439–448. (Google Scholar)

Lo ́pez R. Special Weingarten surfaces foliated by circles, Monatsh. Math. 154 (2008), 289–302. (Google Scholar)

Nitsche J. C. C., Cyclic surfaces of constant mean curvature, Nachr. Akad. Wiss. Gottingen Math. Phys. II 1 (1989), 1–5. (Google Scholar)

Riemann, B. U ̈ber die Fla ̈chen vom kleinsten Inhalt bei gegebener Begrenzung, Abh. K ̈onigl Ges. d. Wissensch. G ̈ottingen, Mathema. C1, 13 (1868), 329–333. (Google Scholar)


Refbacks

  • There are currently no refbacks.


ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr