Definition:Real Submanifold
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Definition
Let $n,k\geq1$ be natural numbers.
Let $M\subset\R^n$ be a subset.
Definition Using Local Diffeomorphisms
$M$ is a real $C^k$-submanifold of dimension $d$ of $\R^n$ if and only if for all $p\in M$ there exists a open neighborhood $U$ of $p$ in $\R^n$ and a differentiable function $\phi : U \to \R^n$ that is a $C^k$-diffeomorphism on its image, such that:
- $\phi(M \cap U) = \phi(U) \cap (\R^d\times\{0\})$
Definition Using Local Submersions
$M$ is a real $C^\infty$-submanifold of dimension $d$ of $\R^n$ if and only if for all $p\in M$ there exists a open neighborhood $U$ of $p$ in $\R^n$ and a $C^\infty$-submersion $\phi : U \to \R^{n-d}$ such that:
- $M \cap U = \phi^{-1}(0)$
Definition Using Local Embeddings
$M$ is a real $C^\infty$-submanifold of dimension $d$ of $\R^n$ if and only if for all $p\in M$ there exists an open neighborhood $U$ in $\R^d$ and a $C^\infty$-embedding $\phi : U \to \R^{n}$ such that:
- $p \in \phi(U) \subset M$