On Integral Invariants and Dralls for a Dual Lorentzian Closed Strip

Abstract

In this paper, a dual Lorentzian closed strip {X˜(s),N(s)} is considered within the context of Lorentzian 3-space, with a real parameter s, a timelike moving frame {A(s),G(s),N(s)} is defined, which moves along the curve of this dual Lorentzian closed strip. A timelike vector D, which is fixed in this frame, is investigated alongside the dual integral invariants (dual angles of pitch, dual pitch) and dralls of the dual Lorentzian closed strip corresponding to the dual Lorentzian closed curve traced by vectors A,G,N and D. Some fundamental relations among these integral invariants of the dual Lorentzian closed strip are studied. Furthermore, these dual results are mapped to the line-space R13 and several theorems are presented regarding the dual Lorentzian geodesic curvature Kg, dual Lorentzian normal curvature Kn and dual Lorentzian geodesic torsion Tg of the strip, accounting for the causal characters (timelike) inherent in Lorentzian geometry.