---
tag: 6RW68JGBHC
kind: concept
name: Immersion
msc: [57R42, 58A05]
sources:
  - url: https://en.wikipedia.org/wiki/Immersion_(mathematics)
same_as:
  wikidata: Q574523
  wikipedia: Immersion (mathematics)
---

## Summary

An immersion is a smooth map between [[smooth manifolds|F45VQ47GPV]] whose [[differential|6P9PXQDSEN]] is injective at every point.
Locally it looks like the inclusion of $\R^m$ into $\R^n$, but globally it may cross itself.
Immersions are the maps along which a manifold can be drawn in another one without corners or folds.
Their classification up to regular homotopy is a model case of the homotopy principle.

## Explanations

### L1 human

Think of drawing a curve on paper with a pen that never stops moving.
The drawing may cross itself, but it never has a corner or a point where the pen stands still.
That is an immersion of a curve into the plane.
For example, the figure eight $t \mapsto (\sin 2t, \sin t)$ immerses the circle into $\R^2$: its velocity is never zero, but the image crosses itself at the origin.
An immersion that is one-to-one and a homeomorphism onto its image is called an embedding.