I know there are a couple of questions on this, but they don't have a fully worked out answer and I'm still trying to wrap my mind around submanifolds so I don't really get what exactly I need to do.
Here is the question:
Prove that the $M=\{(x,y,z)|x^2+y^2-z^2=0 \quad and\quad z\geq0)\}\in\mathbb{R}^3$ is not a $C^1$-submanifold on $\mathbb{R}^3$
Attempt at solution:
For $M$ to be a $k$-dimensional submanifold, we need that for every point in $M$, and therefore also for $(0,0,0)$, there exists an neighborhood $W$ in $\mathbb{R}^3$ and a $C^1$-map $F:W\to\mathbb{R}^{3-k}$ such that
- $\text{rank } dF_r = n-k\quad r\in M\cap W$
- $M\cap W = F^{-1}(0)\cap W$
So the problem is that the rank of the tangent space at the vertex is always going to be $n$, in this case $3$, thus violating the 1st condition. But how do I write this down as a rigorous answer?