Singularities of tangent surfaces to directed curves
arXiv:1607.08702
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.
This paper is a second half of the unpublished paper arXiv:1501.07341 [math.DG], which is divided into two shorter papers, the paper arXiv:1602.02458 [math.DG] (to appear in Journal of Geometry) and the present paper