Ali UÇUM

Canal Surface Whose Center Curve is a Hyperbolic Curve with Hyperbolic Frame

In this paper, we obtain the parametrization of the canal surfaces whose center curves are the hyperbolic curves on the hyperbolic space $H^{2}$ in $%\mathbb{E}_{1}^{3}$. The parametrization of the canal surface is expressed according to the hyperbolic frame given in [10]. Then, the parallelsurface of this surface is studied. Also, we define the notion of the associated canal surface. Lastly, we give the geometric properties of thesesurfaces such that Weingarten surface, $\left( X,Y\right) $-Weingarten surface and linear Weingarten surface.

Keywords:

Canal surfaces, tubular surfaces, hyperbolic curve, Weingarten surface, (X Y )-Weingarten surface,

PDF

___

[1] Chen B.-Y.: Geometry of Submanifolds. Dekker, New York (1973).
[2] Gök İ.: Quaternionic approach of canal surfaces constructed by some new ideas. Adv. Appl. Clifford Algebras. 27 (2), 1175-1190 (2017).
[3] İlarslan K. and Boyacıoğlu Ö.: Position vectors of a spacelike W-curve in Minkowski space E₁³, Bulletin of the Korean Mathematical Society. 44 (3), 429-438 (2007).
[4] İlarslan K. and Boyacıoğlu Ö.: Position vectors of a timelike and a null helix in Minkowski 3-space. Chaos, Solitons & Fractals, 38 (5), 1383–1389 (2008).
[5] Karacan M. K., Es H. and Yaylı, Y.: Singular points of the Tubular Surfaces in Minkowski 3-space. Sarajevo Journal of Mathematics. 14, 73–82 (2006).
[6] Karacan M. K., Yoon D.W. and Tuncer Y.: Tubular Surfaces of Weingarten Types in Minkowski 3-space. Gen. Math. Notes. 22, 44–56 (2014).
[7] Karacan M. K. and Bukcu B.: An alternative moving frame for a tubular surface around a spacelike curve with a spacelike normal in Minkowski 3-space. Rendiconti del Circolo Matematico di Palermo.57, 193–201 (2008).
[8] Karacan M. K. and Tuncer Y.: Tubular Surfaces of Weingarten types in Galilean and Pseudo-Galilean. Bulletin of Mathematical Analysis and Applications. 5, 87–100 (2013).
[9] Kocakuşaklı E., Tuncer O. O., Gök İ. and YaylıY.: A new representation of canal surfaces with split quaternions in Minkowski 3-space. Adv. Appl. Clifford Algebras. 27 (2), 1387–1409 (2017).
[10] Liu H.: Curves in three dimensional Riemannian space forms. Results. Math. 66, 469–480 (2014).
[11] Petrović-Torgašev M. and Šućurović E.: W-curves in Minkowski space-time. Novi Sad J. Math. 32 (2), 55-65 (2002).
[12] Ro J. S. and Yoon D. W.: Tubes of Weingarten types in a Euclidean 3-space. Journal of the Chungcheong mathematical society. 22 (3), 359-366 (2009).
[13] Uçum A., İlarslan, K.: New types of canal surfaces in Minkowski 3-space. Adv. Appl. Clifford Algebras. 26, 449–468 (2016).
[14] Yoon D. W.: On non-developable ruled surfaces in Euclidean 3-spaces. Indian Journal of Pure and Applied Mathematics. 38, 281–290 (2007).