---
tag: S4543S2GBJ
kind: definition
name: Submanifold
msc: [58A05, 57R40]
defines: VKFW27WXEZ
uses: [F45VQ47GPV, J7P32JQTVE, 6RW68JGBHC]
sources:
  - url: https://en.wikipedia.org/wiki/Submanifold
---

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, and let $0 \le k \le n$.
A subset $S \subseteq M$ is an *embedded submanifold* of dimension $k$ if the following holds.
For every point $p \in S$ there is a chart $(U, \varphi)$ of the smooth [[atlas|J7P32JQTVE]] of $M$ with $p \in U$ such that
$$\varphi(U \cap S) = \varphi(U) \cap (\R^k \times \set{0}).$$
Such a chart is a *slice chart* for $S$.
The restrictions of the slice charts give $S$ a unique smooth structure for which the inclusion $S \to M$ is a smooth embedding.

## Remarks

Example: the unit circle $S^1 \subseteq \R^2$ is an embedded submanifold of dimension $1$.
Near a point with $y > 0$, the chart $(x, y) \mapsto (x, y - \sqrt{1 - x^2})$ on $\set{(x, y) : \abs{x} < 1, y > 0}$ sends the circle onto the axis $\R \times \set{0}$; the other points are handled in the same way.
Non-example: the cross $C = \set{(x, y) \in \R^2 : xy = 0}$ is not an embedded submanifold.
Removing the origin from a small connected neighbourhood of $0$ in $C$ leaves four components.
Removing a point from an open interval leaves two, so no slice chart exists at the origin.
The embedded submanifolds of $M$ are exactly the images of smooth embeddings into $M$.
An *immersed submanifold* is the image of an injective [[immersion|6RW68JGBHC]] $S \to M$, with the topology and smooth structure carried over from $S$; its topology may be finer than the subspace topology.