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> [!definition] Definition. ([[embedded submanifold]])
> Let $M$ be a [[smooth manifold]] and $N \subset M$ such that $N$ is itself a [[smooth manifold]]. We say $N$ is an **embedded submanifold of $M$** if
> 1. The [[inclusion map]] $\iota :N \to M$ is [[smooth maps between manifolds|smooth]]
> 2. The [[differential of a smooth map between smooth manifolds|differential]] $(d \iota)_{p}$ is [[injection|injective]] for all $p \in M$ (an [[smooth immersion|immersion]])
> 3. $\iota$ is a [[topological embedding]], i.e., a [[homeomorphism]] onto its image
>
> [!basicexample]
> - The term [[differentiable Euclidean submanifold]] means "embedded submanifold of $\mathbb{R}^{n}$ of class $C^{r}$, $r>1
quot;.
^basic-example
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####
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#### References
> [!backlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM [[]]
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```
> [!frontlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM outgoing([[]])
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```