Helicoidal Surfaces in Some Conformally Flat Pseudo-Spaces of Dimensional Three

Abstract

We introduce the complete pseudo-Riemannian manifold E-3(1) with a conformally flat pseudo-metric satisfying Einstein's equation. First, we get a theorem that associates the curvatures of a non- degenerate surface belonging to conformally equivalent spaces, say E-3(1) and R-1(3). Next, we evaluate the existence of non-degenerate helicoidal surfaces in some conformally flat pseudo-spaces with pseudo-metrics corresponding to particular conformal factors (for example, a certain rotational symmetry with translational symmetry). We determine these conformal factors according to the causal characters of the axis of rotation. Right after, we get a two-parameter family of non-degenerate helicoidal surfaces with prescribed extrinsic curvature or mean curvature given by smooth functions, corresponding to both spacelike and timelike axes of rotation. As for the lightlike axis, we discuss some particular cases.