[[Differential geometry MOC]]
# Real embedded manifold

In these notes, a **real embedded manifold** typically refers to a $C^\infty$ [[submanifold]] of [[real coördinate space]]. #m/def/geo/diff 
The [[Whitney embedding theorem]] provides a sense in which every real [[differentiable manifold]] may be regarded as a real submanifold.

A subset $X \sube \mathbb{R}^N$ is an $n$-dimensional **real submanifold** iff has [[Coördinate chart|charts]] that are $C^\infty$ [[Differentiability#Arbitrary subsets of real coördinate space|diffeomorphisms as subsets of real coördinate space]],
i.e. for every $x \in X$
there exists a neighbourhood $U'$ of $x$ in $\mathbb{R}^N$
and an open set $V \sube \mathbb{R}^n$
such that there exists a $C^\infty$ map $\varphi : U' \twoheadrightarrow V$
with a $C^\infty$ right-inverse $\psi = (\varphi \restriction U' \cap X)^{-1}$.

![[real submanifold.png#invertW]]

## Properties

- [[Total derivative]]
- [[Tangent space]]

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