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$\Bbb RP^1 \to \Bbb RP^2$ is an embedding. Show that $\Bbb RP^1$ can't be denoted as an inverse image of a regular value for some smooth map on $\Bbb RP^2$.

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Suppose $f$ is any smooth function from $\mathbb{R}P^2$ to $\mathbb{R}$ which has $\mathbb{R}P^1$ as a level set. We will show that every point of $\mathbb{R}P^1$ is singular.

By changing $f$ to $f + c$ for a constant $c$, we may assume wlog that $\mathbb{R}P^1 = f^{-1}(0)$.

Since $\mathbb{R}P^2 - \mathbb{R}P^1$ is connected and since $f$ is never $0$ on it, $f$ must have constant sign on $\mathbb{R}P^2 - \mathbb{R}P^1$. By replacing $f$ by $-f$ if necessary, we may assume the sign is positive.

But then $0$ is the global minimum of $f$ and so by standard calculus, the derivative of $f$ must vanish at all points in $\mathbb{R}P^1$. In particular, every point in $\mathbb{R}P^1$ is singular.