Classification Results on Surfaces in The Isotropic 3‐Space

Abstract

The isotropic 3-space which is one of the Cayley-Klein spaces is obtained from the Euclidean space by substituting the usual Euclidean distance with the isotropic distance. In the present paper, we give several classifications on the surfaces in with constant relative curvature (analogue of the Gaussian curvature) and constant isotropic mean curvature. In particular, we classify the helicoidal surfaces in with constant curvature and analyze some special curves on these.

The geometry of Gauss map and shape operator in simply isotropic and pseudo-isotropic spaces

“…Then, there exists a 2parameter family of pseudo-isotropic helicoidal surfaces of i-type with K as the Gaussian curvature and whose generating curves α k 0 ,k 1 (s) = (s, 0, z(s)), or α k 0 ,k 1 (s) = (0, s, z(s)), where s > 0 is an arc-length parameter, satisfy On the other hand, the first fundamental of a parabolic revolution surface of i-type is given by Eqs. (4.50) and (4.51): 2 , if α = (x, 0, z) I = −y ′ 2 du 2 − 2by ′ dudt + (a 2 − b 2 )dt 2 , if α = (0, y, z) .…”

“…Z 3 (u, t) = (a t + x(u), b t, c t + ac 1 + bc2 2 t 2 + c 1 t x(u) + z(u)). (IV) Warped translation surfaces (c = (ac 1 + bc 2 ) = 0; (a, b), (c 1 , c 2 ) = (0, 0)):• If α(u) = (x(u), y(u), 0), we have the warped translation surface of ni-type(4.10) Y 4 (u, t) = (a t + x(u), b t + y(u), c 1 t x(u) + c 2 t y(u));• If α(u) = (x(u), 0, z(u)), we have the warped translation surface of i-type(4.11) Z 4 (u, t) = (a t + x(u), b t, c 1 t x(u) + z(u)).…”

Rotational surfaces in isotropic spaces satisfying weingarten conditions

“…The isotropic 3-space I 3 is obtained from the 3-dimensional projective space P (︀ R 3 )︀ with the absolute figure which is an ordered triple (p, l 1 , l 2) , where p is a plane in P (︀ R 3 )︀ and l 1 , l 2 are two complex-conjugate straight lines in p (see [36]). The homogeneous coordinates in P (︀ R 3 )︀ are introduced in such a way that the absolute plane p is given by x 0 = 0 and the absolute lines l 1 , l 2 by x 0 = x 1 + ix 2 = 0, x 0 = x 1 − ix 2 = 0.…”

“…The motivation of the present paper is to study Weingarten surfaces, in particular Weingarten rotational surfaces, in the isotropic 3-space I 3 which is one the CayleyKlein spaces. Most recently, M. E. Aydin [2] classified the helicoidal surfaces in I 3 , which are natural generalizations of the rotational surfaces, with constant curvature and analyzed some special curves on such surfaces.…”

“…Kamenarovic [17,18], B. Pavkovic [29], Z. M. Sipus [36], M.E. Aydin [1,2] and M. K. Karacan et al [19] have studied some classes of surfaces in I 3 .…”