Why cone of revolution is not a submanifold?

Question

I just begin to learn sub-manifold. I want to prove the cone of revolution $$\{(x,y,z) \in \mathbb{R}^3|x^2+y^2−z^2=0)\}$$ is not a submanifold. Why we cannot find a submersion (like $f(x,y,z) =x^2+y^2−z^2$) to express the cone as an equation? I see there may be some problem on $0$ but I cannot express it explicitly and link it to some classical contrary example (like $[0,1]$ not homeomorphic to $(0,1)$).

Answer 1

Let me use $D$ for the cone (which is really a double cone hence the $D$), and consider the point $P = (0,0,0)$. Note that $D-\{P\}$ is disconnected, in fact it has two components $D^+ = \{(x,y,z) \in D \mid z > 0\}$ and $D^- = \{(x,y,z) \in D \mid z < 0\}$. For any neighborhood $U \subset D$ of $P$, there exist a point $Q_1 \in U \cap D^+$ and another point $Q_2 \in U \cap D^-$. But since $Q_1,Q_2$ are in different components of $D-\{P\}$ they must also be in different components of $U-\{P\}$. This proves that $U-\{P\}$ is disconnected, and this is true for any neighborhood $U$ of $P$.

However, in a 2-dimensional manifold, every point $P$ has a neighborhood $U$ homeomorphic to an open disc and hence $U-\{P\}$ is connected.