Tube surfaces in pseudo-Galilean space

Abstract

In this paper, we introduce the tube surfaces in pseudo-Galilean 3-space. We classify the tube surfaces into two types according to the spine curve which generates the tube surface in pseudo-Galilean space. Moreover, we show that both types of tube surfaces are spacelike surfaces. Then, we begin to study the local geometry of the tube surfaces. We obtain the first and second fundamental forms of the tube surfaces. In addition, we investigate the Gauss and mean curvatures of the tube surfaces in pseudo-Galilean space.

On the tubular surfaces in E3

“…and if we admit as cosθ cosβ − sinθ sinβ = cos(θ + β ) = cosγ, (15) sinθ cosβ + cosθ sinβ = sin (θ + β ) = sinγ (16) and…”

“…Recently, the studies on the tubular surfaces are given in [14,15,16,17]. Generally, a tubular surface generated by constructing a tube around a circle is known as a torus.…”

Twisted surfaces with vanishing curvature in Galilean 3-space

“…If a is zero, then K = 0 and we obtain parameterization (3). For f = 0 the surface is not admissible and f g = f g leads to a contradiction or to the previous subcase since then abf = 0 from the constant term in the condition K = 0.…”

“…With regard to the six-parameter group of motions of G 3 , apart from the absolute plane, there exist two classes of planes in G 3 : Euclidean planes which contain f and in which the induced metric is Euclidean and isotropic planes that do not contain f and in which the induced metric is isotropic. Also, there are four types of lines in G 3 : isotropic lines which intersect f , non-isotropic lines which do not intersect f , non-isotropic lines in ω and the absolute line f . In affine coordinates defined by (x 0 :…”

“…There also exists a pseudo-Galilean 3-space G 3 1 which is related to Galilean 3space in a similar way as Minkowski 3-space is related to Euclidean 3-space, see for instance [2,3,13]. Without going into detail here, the pseudo-Galilean scalar product of two vectors a = (x, y, z) and b = (x 1 , y 1 , z 1 ) in G Because of this there exist four types of isotropic vectors a = (0, y, z) in G 3 1 : spacelike ones (if y 2 −z 2 > 0), timelike ones (if y 2 −z 2 < 0) and two types of lightlike ones (if y = ±z). Analougously as in Minkowski 3-space, instead of the trigonometric functions, the hyperbolic functions must be used to describe rotations.…”