On Razzaboni Transformation of Surfaces in Minkowski 3-Space

TL;DR

This paper studies Razzaboni surfaces generated by Bertrand curves in Minkowski 3-space, exploring their geometric properties, defining a transformation, and showing how it maps surfaces with constant curvature Bertrand geodesics to dual surfaces with opposite curvature.

Contribution

It introduces the Razzaboni transformation for surfaces in Minkowski 3-space and proves its duality property relating surfaces with constant curvature Bertrand geodesics.

Findings

Razzaboni surfaces are characterized in Minkowski 3-space.

The Razzaboni transformation maps surfaces with constant curvature Bertrand geodesics to dual surfaces.

Dual surfaces have Bertrand geodesics with opposite constant curvature.

Abstract

In this paper, we investigate the surfaces generated by binormal motion of Bertrand curves, which is called Razzaboni surface, in Minkowski 3-space. We discussed the geometric properties of these surfaces in M^3 according to the character of Bertrand geodesics. Then, we define the Razzaboni transformation for a given Razzaboni surface. In other words, we prove that there exists a dual of Razzaboni surface for each case. Finally, we show that Razzaboni transformation maps the surface M, which has Bertrand geodesics with constant curvature, to the surface M^* whose Bertrand geodesics also have constant curvature with opposite sign.