I have to construct the circle as a submanifold of $\mathbb R^2$. Let $f(x,y)=x^2+y^2$ which is $\mathcal C^\infty $. The circle is given by $$\mathbb S^1=\{(x,y)\mid x^2+y^2=1\}=f^{-1}(1).$$
We have that $$df_p=\left(\frac{\partial f}{\partial x}(p),\frac{\partial f}{\partial y}(p)\right)=2(p_1,p_2)$$
I have a theorem that says that if $Rg(f,x)$ is constant on an open $U\supset f^{-1}(1)$, then $f^{-1}(1)$ is a submanifold. The problem is that $Rg(df_p)= 1$ if $p\neq 0$ and $Rg(df_p)=0$ if $p=0$, and since every open that contain $\mathbb S^1$ contain also $0$, it can't be a submanifold, right ?
So, how can I continue ?