Corpus ID: 119137567

# Differential geometry of invariant surfaces in simply isotropic and pseudo-isotropic spaces

@article{Silva2018DifferentialGO,
title={Differential geometry of invariant surfaces in simply isotropic and pseudo-isotropic spaces},
author={Luiz C. da Silva},
journal={arXiv: Differential Geometry},
year={2018}
}
• L. C. D. Silva
• Published 28 September 2018
• Mathematics
• arXiv: Differential Geometry
We study invariant surfaces generated by one-parameter subgroups of simply and pseudo isotropic rigid motions. Basically, simply and pseudo isotropic geometries are the study of a three-dimensional 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 7 types of invariant surfaces, which then generalizes the study of revolution/helicoidal surfaces in Euclidean and Lorentzian geometries to the context… Expand
3 Citations
Zero mean curvature surfaces in isotropic three-space
• Mathematics
• 2021
We examine the theory of surfaces in the isotropic threespace, with emphases on the surfaces related to the zero mean curvature.
Holomorphic representation of minimal surfaces in simply isotropic space
It is known that minimal surfaces in Euclidean space can be represented in terms of holomorphic functions. For example, we have the well known Weierstrass representation, where part of theExpand
Invariant surfaces with coordinate finite-type Gauss map in simply isotropic space
• Mathematics
• 2020
Abstract We consider the extrinsic geometry of surfaces in simply isotropic space, a three-dimensional space equipped with a rank 2 metric of index zero. Since the metric is degenerate, a surfaceExpand

#### References

SHOWING 1-10 OF 28 REFERENCES
Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces
In this work, we are interested in the differential geometry of curves in the simply isotropic and pseudo-isotropic 3-spaces, which are examples of Cayley-Klein geometries whose absolute figure isExpand
Constant curvature surfaces in a pseudo-isotropic space
• M. Aydin
• Physics, Mathematics
• Tamkang Journal of Mathematics
• 2018
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 ofExpand
Classification results on surfaces in the isotropic 3-space
The isotropic 3-space I^3 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,Expand
Helicoidal Surfaces in the three dimensional simply isotropic space I
• Mathematics
• 2017
In this paper, we classify helicoidal surfaces in the three dimensional simply isotropic space  I₃¹ satisfying some algebraic equations in terms of the coordinate functions and the LaplacianExpand
A generalization of translation surfaces with constant curvature in the isotropic space
In this paper we study the translation surfaces generated by a space curve and a planar curve in the isotropic 3-space $${\mathbb{I}^{3}}$$I3. We completely classify such surfaces inExpand
Curvature analysis and visualization for functions defined on Euclidean spaces or surfaces
• Mathematics, Computer Science
• Comput. Aided Geom. Des.
• 1994
This work presents the central formulae for a curvature analysis of functions defined on surfaces and shows how to use them for visualization purposes and as a mathematical basis for the construction of interpolating or approximating functions on surfaces. Expand
Loxodromes and geodesics on rotational surfaces in a simply isotropic space
A loxodrome is a curve on a parametrized surface that intersects one family of parametric lines at a constant angle. In this paper, we investigate loxodromes on rotational surfaces in theExpand
Linear Weingarten Helicoidal Surfaces in Isotropic Space
• Computer Science, Mathematics
• Symmetry
• 2016
In the present paper, helicoidal surfaces in the three-dimensional isotropic space I 3 are studied and constructed satisfying a linear equation in terms of the Gaussian curvature and the mean curvature of the surface. Expand
The geometry of Gauss map and shape operator in simply isotropic and pseudo-isotropic spaces
In this work, we are interested in the differential geometry of surfaces in simply isotropic $${\mathbb {I}}^3$$I3 and pseudo-isotropic $${\mathbb {I}}_{\mathrm {p}}^3$$Ip3 spaces, which consists ofExpand
$d$-Minimal Surfaces in Three-Dimensional Singular Semi-Euclidean Space $\mathbb{R}^{0,2,1}$
In this paper, we investigate surfaces in singular semi-Euclidean space $\mathbb{R}^{0,2,1}$ endowed with a degenerate metric. We define $d$-minimal surfaces, and give a representation formula ofExpand