---
tag: VKFW27WXEZ
kind: concept
name: Submanifold
msc: [58A05, 57R40]
sources:
  - url: https://en.wikipedia.org/wiki/Submanifold
same_as:
  wikidata: Q125539
  wikipedia: Submanifold
---

## Summary

A submanifold of a [[smooth manifold|F45VQ47GPV]] $M$ is a subset that is itself a smooth manifold in a way compatible with $M$.
The most common kind, an embedded submanifold, looks near each of its points like a flat coordinate subspace $\R^k \subseteq \R^n$.
Level sets of submersions, images of embeddings and graphs of smooth maps are submanifolds.
A wider notion, the immersed submanifold, is the image of an injective [[immersion|6RW68JGBHC]] with its own topology.

## Explanations

### L1 human

A submanifold is a smooth curve, surface, or higher-dimensional piece sitting inside a bigger smooth space.
Seen through a good magnifying glass, it looks like a flat line or plane inside flat space.
The unit circle in the plane and the unit sphere in $\R^3$ are submanifolds.
The union of the two coordinate axes in $\R^2$ is not, because at the origin it looks like a cross, not like a line.