---
tag: DF8CJ1A9MF
kind: concept
name: Morin singularities
msc: [57R45, 58K40, 58K25]
specializes: [1JG3HFWX3T]
sources:
  - url: https://en.wikipedia.org/wiki/Singularity_theory
---

## Summary

The Morin singularities are the stable [[singularities|1JG3HFWX3T]] 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|A9TYDPD5D7]] is $(1, 1, \ldots, 1, 0)$ with $k$ ones.

## Explanations

### L1 human

The simplest way a map can degenerate is to crush exactly one direction.
Among such degenerations, the fold of a sheet of paper is the mildest.
The next is the cusp, where a fold line ends.
In more dimensions there are further steps, such as the swallowtail, each one rarer than the last.
The Morin singularities are this whole ladder, written in simple coordinates.