Definition:Real Submanifold

Definition
Let $n,k\geq1$ be natural numbers.

Let $M\subset\R^n$ be a subset.

Definition 1
$M$ is a real $C^k$-submanifold of dimension $d$ of $\R^n$ 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 2
$M$ is a real $C^\infty$-submanifold of dimension $d$ of $\R^n$ 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 3
$M$ is a real $C^\infty$-submanifold of dimension $d$ of $\R^n$ 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:
 * $p \in \phi(U) \subset M$

Also see

 * Equivalence of Definitions of Real Submanifold