Geometry of curves passing through Whitney umbrella

TL;DR

This paper investigates the geometric properties of curves passing through a Whitney umbrella using Darboux frames, defining invariants related to curvature and torsion, and exploring their geometric implications.

Contribution

It introduces new invariants based on Darboux frames for curves through Whitney umbrellas and analyzes their geometric significance and divergence properties.

Findings

Defined three invariants related to curvature and torsion.

Analyzed the divergence degrees and top-terms of these invariants.

Explored developable surfaces along the curves.

Abstract

We study geometry of curves passing through a Whitney umbrella by using a Darboux frame along it. We define three invariants by using Frenet-Serre type formula relating to the geodesic curvature, the normal curvature, and the geodesic torsion. We investigate the degrees of divergence and the top-terms of these invariants and their geometric meanings. We also consider a developable surface along the curve.