Singularities of tangent surfaces to directed curves

TL;DR

This paper classifies the local singularities of tangent surfaces to directed curves, revealing that swallowtails and open swallowtails are generic features in these classifications.

Contribution

It completes the local diffeomorphism classification for generic directed curves' tangent surfaces, identifying key singularities like swallowtails.

Findings

Swallowtails appear as generic singularities.

Open swallowtails are also generically present.

Classification is complete for local diffeomorphisms.

Abstract

A directed curve is a possibly singular curve with well-defined tangent lines along the curve. Then the tangent surface to a directed curve is naturally defined as the ruled surface by tangent geodesics to the curve, whenever any affine connection is endowed with the ambient space. In this paper the local diffeomorphism classification is completed for generic directed curves. Then it turns out that the swallowtails and open swallowtails appear generically for the classification on singularities of tangent surfaces.