The Darboux trihedrons of timelike surfaces in the Lorentzian 3-space

Abstract

The parameter curves (c(1)) and (c(2)) intersecting under the angle theta on a timelike surface and any curve (c) passing through the same point with these curves have six different cases according to their causal characters. In this study, the Darboux trihedrons of these three curves and the Darboux instantaneous rotation vectors of these trihedrons are separately calculated for each of these six cases, and the relationships between these vectors are given. In addition, some theorems and results have been obtained on this surface. Besides, the special cases where the parameter curves (c(1)) and (c(2)) intersect perpendicularly are examined and these cases are compared with previous studies.