SECTIONAL CURVATURE OF TIMELIKE RULED SURFACE PART I: LORENTZIAN BELTRAMI-EULER FORMULA

Abstract

In this paper, the Lorentzian version of Beltrami-Euler formula is investigated in R(1)(n). Initially, the first fundamental form and the metric coefficients of generalized timelike ruled surface are calculated and by the help of the Christoffel Symbols, Riemann-Christoffel curvatures are obtained. Thus, the curvatures of spacelike and timelike tangential sections of generalized timelike ruled surface with timelike generating space and central ruled surface are found to be related to the determinant of the first fundamental form of the surface. In addition to this, the relation between the sectional curvature and the distribution parameter of this ruled surface is obtained. Finally, paying attention to the spacelike and timelike central ruled surface of the generalized timelike ruled surface one by one, four different types of Lorentzian Beltrami-Euler formulas are constituted for generalized timelike ruled surface with timelike generating space.