Generic bifurcations of framed curves in a space form and their envelopes

TL;DR

This paper classifies the generic singularities and bifurcations of framed curves in various space forms using projective geometry and differential systems, providing a comprehensive understanding of their geometric behavior.

Contribution

It introduces a unified classification of singularities and bifurcations for framed curves in Euclidean, elliptic, and hyperbolic spaces using projective geometric methods.

Findings

Classified singularities for framed curves in different space forms.

Identified bifurcation types for one-parameter families of curves.

Used differential systems on flag manifolds to analyze frames.

Abstract

The generic singularities and bifurcations are classified for one-parameter families of curves with frames in a space form, the Euclidean space, the elliptic space or the hyperbolic space via projective geometry. Two kinds of frames are considered, adapted frames and osculating frames, in terms of certain differential systems on flag manifolds.