Constant curvature surfaces in a pseudo-isotropic space

TL;DR

This paper investigates the local geometry of curves and surfaces in a pseudo-isotropic space, deriving curvature formulas and classifying surfaces with constant Gaussian and mean curvature, including surfaces of revolution.

Contribution

It introduces curvature formulas for curves and surfaces in pseudo-isotropic space and characterizes surfaces of revolution with constant curvatures.

Findings

Formulas for curvature, torsion, and Frenet frames in pseudo-isotropic space.

Classification of timelike surfaces with constant Gaussian and mean curvature.

Description of surfaces of revolution generated by hyperbolic rotations.

Abstract

In this study, we deal with the local structure of curves and surfaces immersed in a pseudo-isotropic space I_{p}^{3} that is a particular Cayley-Klein space. We provide the formulas of curvature, torsion and Frenet trihedron in order for spacelike and timelike curves. The causal character of all admissible surfaces in I_{p}^{3} has to be timelike or lightlike up to its absolute. We introduce the formulas of Gaussian and mean curvature for timelike surfaces in 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.