Embedded submanifold

Let $M$ be a smooth manifold with dimension $n$. A subset $S\subset M$ is a submanifold (of dimension $k$) of $M$ if for every point $x\in S$ there exists a chart $\varphi :U\subset S\subset M\to W\subset {\mathbb{R}}^n$ such that $x\in U$ and $\varphi (U\cap S)=\{(x_1,\dots ,x_n)\in \varphi (U)|x^{k+1}=\cdots =x^n=0\}=W\cap (\{0\}\times {\mathbb{R}}^k)$

This will definitely have a smooth manifold structure since we can take $U\cap S$ to be the charts on $S$ which are already smooth charts. We can “forget” about the extra dimensions since in $S$ they are all 0 anyway.

Alternative definition: (from @lee2013) We could also say that a submanifold is a subset $S\subset M$ that is a manifold such that the inclusion map $S\hookrightarrow M$ is a smooth embedding.

Tangent space of submanifolds

Main point: If $S\subset M$ is a submanifold, then for $p\in S$, $T_{p}S\subset T_{p}M$ is a linear subspace.

Since $\iota :S\to M$ is a smooth immersion by definition, then $d{\iota }_p(v)$ is injective for $v\in T_{p}S$. We can see this directly using the definition of the differential: For $\tilde{v}=d{\iota }_p(v)\in T_{p}M$

$\tilde{v}=d{\iota }_p(v)=v(f\circ \iota )=v(f{|}_S)$

So in order to evaluate a tangent vector at a point, we just restrict the function to the submanifold.

Characterization of the submanifold tangent space

For a submanifold $S\subset M$, $T_{p}S\subset T_{p}M$ can be characterized as the subspace such that $T_{p}S=\{v\in T_{p}M|v(f)=0\text{for }f\in C^{\infty }(M)\text{such that }f{|}_S=0\}$

We can also use a defining map to identify the tangent space of the submanifold.