Singularities of Blaschke normal maps of convex surfaces

TL;DR

This paper establishes a relationship between the singularities of the Blaschke normal map of convex surfaces and the Euler number of a specific subset related to the affine shape operator, revealing new geometric insights.

Contribution

It provides a novel link between singularity counts of the Blaschke normal map and topological invariants of convex surfaces in affine geometry.

Findings

Difference in positive and negative swallowtail counts equals the Euler number of a certain subset.

The result connects singularity theory with affine differential geometry.

Offers a new perspective on the structure of convex surfaces in affine space.

Abstract

We prove that the difference between the numbers of positive swallowtails and negative swallowtails of the Blaschke normal map for a given convex surface in affine space is equal to the Euler number of the subset where the affine shape operator has negative determinant.