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Let $f:\mathbb{R}^n\to \mathbb{R}$ be a smooth function. Then for any regular value $a\in \mathbb{R}$ of $f$, its level set $f^{-1}(a)$ is orientable: the gradient of $f$ is nowhere tangent to $f^{-1}(a)$, so it determines an orientation on $f^{-1}(a)$. Now suppose $a$ is a critical value of $f$. Then $f^{-1}(a)$ is may not even be a manifold, but however let us assume that it is an embedded submanifold of $\mathbb{R}^n$. Is there such an example with $f^{-1}(a)$ nonorientable? (I've seen that a closed hypersurface of $\mathbb{R}^n$ is always orientable, so $f^{-1}(a)$ must not be compact.)

Or, more generally, if $f:M\to\mathbb{R}$ is a smooth function on a smooth manifold, can $f^{-1}(a)$ with $a$ a critical value of $f$, be a nonorientable embedded submanifold?

Edit. As in the comment, this is true, by considering an embedding $g$ of $\mathbb{RP}^2$ in $\mathbb{R}^4$, and choosing a smooth function $f:\mathbb{R}^4\to \mathbb{R}$ such that $f^{-1}(0)$ is the image of $g$.

Now I'm curious, for $f:\mathbb{R}^n\to \mathbb{R}$ a smooth function, can $f^{-1}(a)$ with $a$ a critical value of $f$, be a nonorientable embedded hypersurface?

blancket
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    It was discussed many times on this site that every closed subset of $\mathbb R^n$ is the zero level set of some smooth function. Can you conclude now? – Moishe Kohan May 24 '25 at 02:33
  • @MoisheKohan Thanks, but can a noncompact, nonorientable manifold be embedded as a hypersurface in $\mathbb{R}^n$? – blancket May 24 '25 at 03:10
  • I did not say "hypersurface" and you did not ask for one. – Moishe Kohan May 24 '25 at 03:36
  • @MoisheKohan I see. I should consider $\mathbb{RP}^2$ embedded in $\mathbb{R}^4$, then. By the way, I'm still curious that whether a noncompact nonorietnable manifold can be embedded as a hypersurface in $\mathbb{R}^n$ or not. – blancket May 24 '25 at 03:51
  • Yes, say Moebius band in $R^3$. But it cannot be a closed subset. – Moishe Kohan May 24 '25 at 04:00
  • Your last question is already answered in the linked post (read George's answer and clarifying comments). – Moishe Kohan May 30 '25 at 14:48

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