I have the following definition:
Definition: $M\subset\mathbb R^{n+k}$ is a $n$-dimensional submanifold of class $C^{p}$ of $\mathbb R^{n+k}$ if for every $x\in M$ there is a neighbourhood $U$ of $x$ in $\mathbb R^{n+k}$ and a submersion $f:U\longrightarrow \mathbb R^k$ of class $C^p$ such that $U\cap M=f^{-1}(0)$.
Recall $f$ is a submersion if $df_x$ is surjective for all $x$.
I'm trying to use this definition to show $$S^n=\{x\in\mathbb R^{n+1}: |x|=1\}$$ is a $C^\infty$ $n$-dimensional submanifold of $\mathbb R^{n+1}$.
Up to now: For every $x\in S^n$ taking $U=\mathbb R^{n+1}$ we have $U$ is a neighbourhood of $x$ in $\mathbb R^{n+1}$. Furhtermore, the map $f:U\longrightarrow \mathbb R$ given by $$f(x)=|x|^2-1$$ is $C^\infty$ in $U$. In fact, $1\in C^\infty(U)$ and (using multi-index notation) $$\displaystyle |x|^2=\sum_{|\alpha|\leq 2} c_\alpha x^\alpha$$ for some $c_\alpha\in\mathbb R$, that is, $|x|^2$ is a polynomial in $\mathbb R^{n+1}$ so that $|x|^2\in C^\infty(U)$.
I should be able to show $f$ is a submersion. But if I'm not wrong $$df_x(v)=2\langle x, v\rangle$$ and this is not surjective for $x=0$. Did I miss something?
Finally, it is clear $U\cap S^n=S^n=f^{-1}(0)$.
The only problem is whether or not $f$ is a submersion.
Right, $f$ is not a submersion. Now if you choose a different $U$...