Spacelike surface geometry

Abstract

In this paper, by considering v = constant and u = constant parameter curves on spacelike surface x = x(u, v), (c(1)) and (c(2)), respectively, and any spacelike curve (c) 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:

omega(1) = (1/(T-g)(1) + cos theta/sin theta(R-n)(1))t(1) - 1/sin theta(R-n)(1)t(2) - 1/(R-g)(1)N,

omega(2) = 1/sin theta(R-n)(2)t(1) + (1/(T-g)(2) - cos theta/sin theta(R-n)(2))t(2) - 1/(R-g)(2)N,

omega = 1/sin theta (sin(theta-phi)/Tg + cos(theta-phi)/R-n)t(1) + 1/sin theta (sin phi/T-g - cos phi/R-n)t(2) - 1/RgN

and we have obtained the relationship between these vectors as

omega = sin(theta-phi)/sin theta omega(1) + sin phi/sin theta omega(2) -d phi/dsN,

where theta and phi are the spacelike angles between tangent vectors of (c(1)) and (c(2)) curves, and of (c) and (c(1)) curves, respectively. N 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.