---
tag: 1JG3HFWX3T
kind: concept
name: Singularity of a smooth map
msc: [58K05, 57R45, 58K15]
sources:
  - url: https://en.wikipedia.org/wiki/Singularity_theory
---

## Summary

A singularity, or singular point, of a smooth map between [[smooth manifolds|F45VQ47GPV]] is a point where the [[differential|6P9PXQDSEN]] has less than the largest possible rank.
At all other points the map is locally a linear projection or a linear inclusion, by the rank theorem.
So the singular points are the only places where the local shape of a map can be interesting.
Their study and classification is the subject of [[singularity theory|42V5XGZS4F]] of smooth maps.

## Explanations

### L1 human

Think of a smooth map as a way of projecting, folding or pressing one space onto another.
Where the map is regular, it looks like a flat projection, with nothing special happening nearby.
A singular point is a place where the map folds, pinches or crushes directions together.
For example, the map $x \mapsto x^2$ from the line to the line folds the line in half at $0$, and $0$ is its only singular point.
Shadows of curved objects show the same thing: the outline of the shadow is the image of the singular points of the projection.