Helicoidal Surfaces in the three dimensional simply isotropic space I₃¹

Karacan, Murat Kemal; Yoon, Dae Won; Kızıltuğ, Sezai

doi:10.5556/j.tkjm.48.2017.2200

Cited by 9 publications

(6 citation statements)

References 9 publications

(3 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…After listing all invariant surfaces, which are divided into 7 basic types, we compute their mean and Gaussian curvatures and we also solve the problem of prescribed curvatures for the so-called invariant surfaces of i-type (see Definition 1; for the ni-type we solve the prescribed Gaussian curvature problem for helicoidal surfaces only). These findings generalize the study of helicoidal surfaces in I 3 [2,9] and revolution surfaces in I 3 p [3] with constant curvatures. This work is divided as follows.…”

Section: Introductionsupporting

confidence: 83%

“…The study of I 3 has been initiated by the Austrian geometer Karl Strubecker in the 1930's [17,18,19,20,21] (see also [14] and references therein), while that of I 3 p began only recently [3,7]. Besides its mathematical interest [1,3,9,23,24], see also the recent contributions by this Author [6,7], isotropic geometry finds applications in economics [4,5], image processing [10], and shape interrogation [12]. In addition, this theory may prove useful in understanding the geometry of surfaces with zero mean curvature in semi-Riemannian spaces [15,16].…”

Section: Introductionmentioning

confidence: 99%

“…2 of [14]. However, in I 3 much attention has been paid only to helicoidal and revolution surfaces, [23] and [2,9,24], respectively, while revolution surfaces in I 3 p were studied in [3]. Here we revisit the 1-parameter subgroups of simply isotropic rigid motions by exploiting their linear representation in the group of invertible matrices GL(4, R), which we believe offers the advantage of being simpler and easier to follow.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Silva

2019

J. Geom.

View full text Add to dashboard Cite

We study invariant surfaces generated by one-parameter subgroups of simply and pseudo isotropic rigid motions. Basically, the simply and pseudo isotropic geometries are the study of a threedimensional space equipped with a rank 2 metric of index zero and one, respectively. We show that the one-parameter subgroups of isotropic rigid motions lead to seven types of invariant surfaces, which then generalizes the study of revolution and helicoidal surfaces in Euclidean and Lorentzian spaces to the context of singular metrics. After computing the two fundamental forms of these surfaces and their Gaussian and mean curvatures, we solve the corresponding problem of prescribed curvature for invariant surfaces whose generating curves lie on a plane containing the degenerated direction.Mathematics Subject Classification. Primary 53A35; Secondary 53A10; 53A40.

show abstract

Section: Introductionsupporting

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Silva

2019

J. Geom.

View full text Add to dashboard Cite

show abstract

“…Intending a similar approach for the isotropic geometry, we are interested in the local theory of curves and surfaces in I 3 p . For details of isotropic geometry, see [1]- [4], [12,15], [23]- [25], [31].…”

Section: Introductionmentioning

confidence: 99%

Constant curvature surfaces in a pseudo-isotropic space

Aydın¹

2018

Tamkang J. Math.

View full text Add to dashboard Cite

In this study, we deal with the local structure of curves and surfaces immersed in a pseudo-isotropic space $\mathbb{I}_{p}^{3}$ that is a particular Cayley-Klein space. We provide the formulas of curvature, torsion and Frenet trihedron for spacelike and timelike curves, respectively. The causal character of all admissible surfaces in $\mathbb{I}_{p}^{3}$ has to be timelike up to its absolute. We introduce the formulas of Gaussian and mean curvature for timelike surfaces in $\mathbb{I}_{p}^{3}$. As applications, we describe the surfaces of revolution which are the orbits of a plane curve under a hyperbolic rotation with constant Gaussian and mean curvature.

show abstract

“…where J = I, II and i = 1, 2, 3, in these spaces in [20,21] and [22], respectively. Also, in [23], [24] and [25], authors studied affine translation surfaces, helicoidal surfaces and ruled surfaces satisfying the same condition.…”

Section: Introductionmentioning

confidence: 99%

Warped Translation Surfaces of Finite Type in Simply Isotropic 3-Spaces

Kelleci

2020

Fundamental Journal of Mathematics and Applications

View full text Add to dashboard Cite

In this paper, we classify warped translation surfaces being invariant surfaces of i-type, that is, the generating curve has formed by the intersection of the surface with the isotropic xz-plane in the three-dimensional simply isotropic space I 3 under the conditionHere, ∆ J is the Laplace operator with respect to first and second fundamental form and λ i , i = 1, 2, 3 are some real numbers. Also, as an application, we give some examples for these surfaces and also some explicit graphics of them. All graphics have been plotted with Maple14.

show abstract

Helicoidal Surfaces in the three dimensional simply isotropic space I₃¹

Cited by 9 publications

References 9 publications

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

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

Constant curvature surfaces in a pseudo-isotropic space

Warped Translation Surfaces of Finite Type in Simply Isotropic 3-Spaces

Contact Info

Product

Resources

About