Morin singularities

Summary

The Morin singularities are the stable singularities of smooth maps whose differential drops rank by exactly one. For maps between manifolds of the same dimension $n$ they form one sequence, with one member for each order $k$ with $1 \le k \le n$. It begins with the fold, the cusp and the swallowtail. Morin gave their normal forms, and they are exactly the generic corank one singularities. Their Thom–Boardman symbol is $(1, 1, \ldots, 1, 0)$ with $k$ ones.