$\bf 1.7$ Submersions. Quotient Manifolds
Problem $\bf1.7.1\;$ Let $f:\mathbb R^3\to\mathbb R$ be given by $f(x,y,z)=x^2+y^2-1.$
$\quad(1)$ Prove that $C=f ^{-1}(0)$ is an embedded $2$-submanifold of $\mathbb R^3.$
Can you help for solving this problem ı am triying to understand embedded 2 submanifold please help me.
Clearly any point on $C$ looks like $ (\cos(\theta),\sin(\theta),z ) $, so you can find suitable open sets in $\mathbb{R}^2 $ and map $ \phi : C \rightarrow \mathbb{R}^2 $ as $\phi(\cos(\theta),\sin(\theta),z) = (\theta,z) $ as charts making $C$ a two dimensional submanifold embedded in $\mathbb{R}^3 $ which is infact a right circular cylinder.