# Evolution of Spacelike Curves and Special Timelike Ruled Surfaces in the Minkowski Space

## TL;DR

This paper investigates the evolution of spacelike curves and timelike ruled surfaces in Minkowski space, deriving equations for curvature and torsion, and analyzing surface properties and their applications.

## Contribution

It introduces new evolution equations for spacelike curves and inextensible evolutions of timelike ruled surfaces in Minkowski space, including conditions for inelastic surface evolution.

## Key findings

- Derived time evolution equations for curvature and torsion.
- Computed fundamental forms and curvatures of timelike ruled surfaces.
- Provided exact solutions for the evolution equations.

## Abstract

In this paper, we get the time evolution equations of the curvature and torsion of the evolving spacelike curves in the Minkowski space. Also, we give inextensible evolutions of timelike ruled surfaces that are produced by the timelike normal and spacelike binormal vector fields of spacelike curve and derive the necessary conditions for an inelastic surface evolution. Then, we compute the coefficients of the first and second fundamental forms, the Gauss and mean curvatures for timelike special ruled surfaces. As a result, we give applications of the evolution equations for the curvatures of the curve in terms of the velocities and get the exact solutions for these new equations.