Surfaces of Constant Curvature in the Pseudo-Galilean Space

Željka Milin Šipuš,Blaženka Divjak

doi:10.1155/2012/375264

Abstract

We develop the local theory of surfaces immersed in the pseudo-Galilean space, a special type of Cayley-Klein spaces. We define principal, Gaussian, and mean curvatures. By this, the general setting for study of surfaces of constant curvature in the pseudo-Galilean space is provided. We describe surfaces of revolution of constant curvature. We introduce special local coordinates for surfaces of constant curvature, so-called the Tchebyshev coordinates, and show that the angle between parametric curves satisfies the Klein-Gordon partial differential equation. We determine the Tchebyshev coordinates for surfaces of revolution and construct a surface with constant curvature from a particular solution of the Klein-Gordon equation.

Highlights

Study of differential geometry of curves and surfaces in Euclidean, as well as in other nonEuclidean ambient spaces, has a long history
Classical context of the Euclidean space is a source of results which could be transferred to some other geometries
One way of defining new geometries is through Cayley-Klein spaces

Summary

Introduction

Study of differential geometry of curves and surfaces in Euclidean, as well as in other nonEuclidean ambient spaces, has a long history. General theory of differential geometry of curves and surfaces in Cayley-Klein spaces can be found in 1. Foundations of these areas in the pseudo-Galilean space were established in 2 , as well as in the papers 3–7. We study the angle between the Tchebyshev curves on a surface of constant curvature and show that the angle satisfies the Klein-Gordon partial differential equation. In this respect, the Klein-Gordon equation plays the analogous role in the pseudo-Galilean space as the sine-Gordon equation in Euclidean space. Similar problem is treated in for the Galilean space and in wider context in by means of Cartan frames

Preliminaries

The Gaussian Curvature of Surfaces in G13

Surfaces of Revolution

Klein-Gordon Equation and the Tchebyshev Coordinates in G13