# Submanifolds as Inverse Images

1. Feb 25, 2008

### Kreizhn

1. The problem statement, all variables and given/known data

Show that there exists no smooth function $f : \mathbb{RP}^2 \rightarrow \mathbb{R} \text{ such that } f^{-1}(q) = \mathbb{RP}^1$ for some regular value q of f.

2. The attempt at a solution

So we can quite simply show that $\mathbb{RP}^1$ is a submanifold of $\mathbb{RP}^2$, which is what the first part of the question actually asked but I have omitted here since I don't think it's important. I've tried looking at a few things, but I have one method that really seems to scream out at me but I'm not sure how to use it.

Theorem: Let $f : M\rightarrow N$ be a smooth manp and $P \subseteq N$ be a submanifold of N. If $f \pitchfork P$, then $f^{-1}(P)$ is a submanifold of M, provided that $f^{-1}(P)\neq \emptyset$.
Here $f\pitchfork P$ means f is transverse to P. I want to use contradiction, but this theorem seems to have the implication in the wrong direction in order for me to fully utilize it. Any ideas?