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I found the following two posts to be interesting when I'm studying Sard's Theorem by myself:

On the converse of Sard's theorem

https://mathoverflow.net/questions/423475/a-modified-version-of-the-converse-to-the-sards-theorem

In particular, is there a way to prove the neat claim in the mathoverflow post (without using advanced tools like triangulation of manifolds)? The questions is restated below:

For any manifold $X$ with $\dim X \geq 1$, there always exists some map $f:X \rightarrow \mathbb{R}^2$, such that the differential $df_{x}$ is nonzero for any $x \in X$.

(Here we don't require $df_{x}$ to be nonsingular for any $x \in X$, which weakens the claim.)

I think the claim should be true after trying several examples, but I haven't found a way to prove it...thanks for any help in advance!

mathisfun
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  • Maybe you could try glueing together maps: Take charts $\phi\colon U_\phi \to \Bbb R^n$ and compose them with projections/inclusions $\Bbb R^n\to \Bbb R^2$. Then glue them together with a partition of unity. But this basically amounts to a triangulation… and im not certain if it works; it was just my first idea after reading your post. – GhostAmarth Jun 01 '22 at 03:39
  • Hi there, thanks a lot for your response! Would you please elaborate a little bit more on how I may glue the compositions via the functions $\rho_i$ in the partition of unity? Also, just to confirm, $n$ here denotes the dimension of $X$, right? – mathisfun Jun 01 '22 at 03:51
  • I gave it some more thought and sadly it doesn't work out as easily as I thought. Sorry. I hope someone else has a good idea... – GhostAmarth Jun 01 '22 at 05:52
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    You can think of $f$ as a pair of functions $f_1,f_2:X\to\mathbb{R}$, and then the problem becomes finding $f_1,f_2$ whose critical points do not coincide. Some results on Morse functions would probably be useful there. – Kajelad Jun 01 '22 at 08:49
  • I see, thanks for the hint! So I guess we probably need to pick two Morse functions and use the fact that nondegenerate critical points are isolated? However, I'm still a bit unsure how to pick these two Morse functions... – mathisfun Jun 01 '22 at 10:13
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    @mathisfun The only result you really need is that functions with isolated critical points exist. Then you can pick any two of them, and if any of their critical points coincide you can show that it's possible to pull them apart with small local modifications. – Kajelad Jun 01 '22 at 10:27
  • Yeah the part I'm getting stuck on is how to make the set of critical points of two Morse functions distinct? Currently what I'm thinking of is to take some function $f$ and two $a,b \in \mathbb{R}^{n} (n = \dim X)$. Then we may construct two Morse functions by $f_a(x) = f(x) +a^{T}x$ and $f_{b}(x) = f(x) +b^{T}x$. Then we have that the set of critical points of the two functions are $(\frac{\partial f}{\partial x})^{-1}(-a)$ and $(\frac{\partial f}{\partial x})^{-1}(b)$, but currently I don't know how to proceed... – mathisfun Jun 01 '22 at 16:57
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    @mathisfun You can take a more local approach: Given any point $x\in M$ and any open neighborhood $U\ni x$, there is a diffeomorphism $\varphi:M\to M$ which fixes $M\setminus U$ such that $\varphi(x)\neq x$. Composing a function $f$ with an isolated critical point at $x$ with such a diffeo will move its critical point, without changing its value outside of $U$. – Kajelad Jun 01 '22 at 18:46
  • @Kajelad: I think you should write this argument as an actual answer in order to complete this discussion. – Moishe Kohan Jun 01 '22 at 22:59

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