Notes on Degenerate Curves in Pseudo-Euclidean Spaces of Index Two

TL;DR

This paper studies special types of curves in pseudo-Euclidean spaces of index two, characterizing their properties and relationships, including Bertrand, evolute-involute correspondence, and pseudo-spherical null curves.

Contribution

It introduces new characterizations of degenerate and null curves, and explores their geometric relationships in pseudo-Euclidean spaces of index two.

Findings

Characterization of Bertrand curves in pseudo-Euclidean spaces.

Establishment of evolute-involute correspondence for null and spacelike curves.

Criteria for pseudo-spherical null curves based on curvature functions.

Abstract

In this paper we deal with curves with degeneration degree two in pseudo-Euclidean spaces of index two. We characterize Bertrand curves. We show a correspondence between the evolute of a null curve and the involute of a certain spacelike curve in the $6-$dimensional pseudo-Euclidean space of index two. Also we characterize pseudo-spherical null curves in the $n-$dimensional pseudo-Euclidean space of index two in terms of the curvature functions.