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

Abstract

In this work, we consider the problem of finding explicit parametrizations for non-degenerate helicoidal surfaces with prescribed curvatures in some conformally flat pseudo-spaces with conformal pseudo-metrics whose conformal factors are related to three types of generic cylindrical functions. In the first two, we get a two-parameter family of these surfaces with prescribed extrinsic curvature or mean curvature given by smooth functions. In the last one, we discover a one-parameter family when both curvatures of these surfaces are zero; however, we find a two-parameter family when either one of those curvatures is zero. Also, we support them with examples.

Keywords

• Helicoidal surface
• extrinsic curvature
• conformally flat pseudo-space
• conformally flat pseudo-metric

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