Is $\{(x,y) \in \Bbb R^2: x \cdot y=1\}$ a submanifold?

Question

Meanwhile it is well know that $$M:=\{(x,y) \in \Bbb R^2: x \cdot y=0\}$$ not a submanifold of $\Bbb R^2$ is. The next question is now, is for example $$L:=\{(x,y) \in \Bbb R^2: x \cdot y=1\}$$ Is now $L$ a submanifold of $\Bbb R^2$?

Answer 1

You can use Submersion theorem to show this. If you don't want to use this theorem, then you must explicitly construct a smooth atlas of this submanifold which should be easy.