The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space R2,1

Abstract

The study of the flows of curves is one of the most fascinating research areas in differential geometry. In this paper, we investigate the geometry of the flows of timelike curves according to the quasi-frame in Minkowski space R2,1 (In this paper, we refer to these curves as “quasi-timelike curves”). We investigate the evolution of quasi-timelike curves using the velocity functions and obtain the necessary and sufficient conditions for inextensibility. Additionally, we obtain the explicit forms of the time evolution equations for the quasi-orthonormal frames (tangent, quasi-normal, and quasi-binormal vectors) of the quasi-timelike curve as well as the time evolution equations of their quasi-curvatures. We present a new application for motion with velocities equal to the quasi-curvatures of the quasi-timelike curve. In this application, the time evolution equations of the quasi-curvatures arise as a system of partial differential equations with the form of the heat equation, and by solving this system, we visualize the evolution of quasi-curvatures and the evolution of the quasi-timelike curve. In addition, the acceleration functions are used to investigate the flows of inextensible quasi-timelike curves, and an application for accelerations equal to the quasi-curvatures is given. Through this application, the position vector of the quasi-timelike curve satisfies the one-dimensional wave equation, and the time evolution equations of the quasi-curvatures arise as a system of transport equations. We obtain the solutions and graph them using Wolfram Mathematica 12.