I am trying to do this question in Bredon's Topology and geometry about using the transversality theorem to show that the intersection of two manifolds is a manifold.
Now it goes as follows:
Let $f(x,y,z)=(2-(x^2+y^2)^{1/2})+z^2$ on $\mathbb{R}^3 - (0,0,z)$. Then one can show that $M=f^{-1}(1)$ is a manifold.
Now let $N = \{ (x,y,z) \in \mathbb{R}^3 | x^2+y^2=4 \}$. I need to show that $M \cap N$ is a manifolds by using the transversality theorem, but I'm not quite sure how to do this.
I thought about finding the tanget planes to each to these manifolds and then showing that there a points in one of them that aren't in the other, but I dont know if this will work. So I was wondering if I could get some hints on this.
Thank you
I couldn't fit my response in the character limit of the comment box.
No, your first function needs to be the function $f$ you started with :) You're correct that I was guiding you just to use the regular value set-up. Of course, this is a special case of the transversality theorem. If you try to study the function $h(x,y,z)=x^2+y^2$ on $M$, you end up with a messy local-coordinates computation, so I always discourage my students from trying to show "directly" that the function $h\colon M\to\mathbb R$ has $4$ as a regular value. You could try to see that the inclusion map $\iota\colon M\to\mathbb R^3$ is transverse to the cylinder $N$: As you surmised, you just need to see that at any point $P\in M\cap N$, $T_PM+T_PN = \mathbb R^3$. The way I was leading you is doing this geometrically, by showing that $\nabla f(P)$ (which is the normal vector to $M$) and $\nabla h(P)$ (which is the normal vector to $N$) are linearly independent at every $P\in M\cap N$.