[[Differential geometry MOC]]
# Submanifold

Let $N$ be a $n$-dimensional $C^\alpha$ [[differentiable manifold]] and let $1 \leq m \leq n$.
A $m$-dimensional (embedded) $C^\alpha$ **submanifold** $M \leq N$ is a subset
such that every $p \in M$ has a [[coördinate chart]] $(x,U)$ 
such that $x(U \cap M) = x(U) \cap (\mathbb{R}^m \times \{ \vab 0 \})$. #m/def/geo/diff 

![[submanifold-chart.svg#invert]]


Thus $M$ inherits the structure of an $m$-dimensional $C^\alpha$ [[differentiable manifold]] from $N$,
with the induced chart $(\bar  x, U \cap M)$.


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#state/tidy | #lang/en | #SemBr