Geometry on curves which go around a Whitney umbrella

TL;DR

This paper investigates the geometric properties of curves around a Whitney umbrella, analyzing geodesic and normal curvatures, and deriving functions that describe the surface's geometry, especially focusing on asymptotic behavior.

Contribution

It introduces new functions representing geometric features of curves on Whitney umbrellas and studies their behavior near the singularity.

Findings

Derived functions for geodesic and normal curvatures

Analyzed asymptotic behavior of geometric functions

Enhanced understanding of surface geometry near Whitney umbrella singularity

Abstract

We consider curves which go around Whitney umbrella. Then we consider the geodesic and the normal curvatures, ruled surfaces generated by the normal vector and normal developable surfaces with respect to the tangent and bi-tangent vectors of the curve. Then we obtain several functions which represents geometry on Whitney umbrella. Looking at the top terms of them with respect to the radius, we study geometry on Whitney umbrella.