Generalized tubes in pseudo‐Galilean 3‐space: Split semi‐quaternionic representations and an application to magnetic flux tubes

Abstract

This paper deals with generalized tube surfaces (GTs) and their geometric properties in pseudo‐Galilean 3‐space. We classify these surfaces into two types. We firstly compute the first and second fundamental forms to investigate geometric properties of a GT. Then, we obtain the condition for such a surface to be minimal and present some results which express the conditions for which parameter curves on a GT are geodesics, asymptotics, or lines of curvature. Furthermore, we show how to form GTs by using split semi‐quaternions or their matrix representations. Finally, as an application, we introduce generalized magnetic flux tubes in pseudo‐Galilean 3‐space and obtain the local magnetic field components of such surfaces. The theory studied in the paper is supported by illustrated examples.

Representation of generalized circular surfaces as rigid body motions

“…[2,3,5,6,9,23,26,27]. Similar problem has been also considered in Minkowski and (pseudo-) Galilean spaces by using split quaternions [4,13,21] and (split-) semi quaternions [28].…”