Spacelike surface geometry

Abstract

In this paper, by considering [Formula: see text] and [Formula: see text] parameter curves on spacelike surface [Formula: see text], [Formula: see text] and [Formula: see text], respectively, and any spacelike curve [Formula: see text] that passes through the intersection point of these parameter curves, we have found the Darboux instantaneous rotation vectors of Darboux trihedrons of these three curves, as follows: [Formula: see text] [Formula: see text] [Formula: see text] and we have obtained the relationship between these vectors as [Formula: see text] where [Formula: see text] and [Formula: see text] are the spacelike angles between tangent vectors of [Formula: see text] and [Formula: see text] curves, and of [Formula: see text] and [Formula: see text] curves, respectively. [Formula: see text] is the unit normal vector of the surface. Besides, we have given Euler, Liouville, Bonnet formulas and Gauss curvature of the spacelike surface with new statement.