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I apologize in advance for asking such an easy question here.

Assume that I have a compact smooth manifold $X$ together with a $C^\infty$ function $f:\mathbb{R}\times X\to\mathbb{R}$ such that for every $s\in\mathbb{R}$, $f_s$ is a submersive at any $x\in f_s^{-1}(0)$.

I would like to prove that the $f_s^{-1}(-\infty,0)$ are all diffeomorphic ($s\in\mathbb{R}$).

It sounds like I should use Morse theory, but I haven't been able to figure out how !

In order to use Morse theory one would probably like to find a function $g:X\to \mathbb{R}$ such that $\{g(x)<s\}=\{f(s,x)<0\}$. If it was a local question one could use implicit function theorem. I am probably just being blind...

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    $\begingroup$ Is $X$ compact? I am picturing $X$ a punctured plane and $f(s,x,y)=x-s$. $\endgroup$
    – Ben McKay
    Sep 10, 2013 at 20:19
  • $\begingroup$ Yes. I should have said that. Sorry. $\endgroup$
    – teach
    Sep 10, 2013 at 22:02
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    $\begingroup$ You adapt a proof in Milnor's book on Morse theory. Use the normalized gradient flow to build your diffeomorphism. $\endgroup$
    – Charlie Frohman
    Sep 11, 2013 at 2:00
  • $\begingroup$ Apply Morse theorem to $Y=f^{-1}(0)$ being a submanifold of $\mathbb R\times X$, and the function $g(s,x)=s$ restricted to $Y$. This function on $Y$ is proper and regular on, hence Morse theory applies. $\endgroup$
    – Sergei Ivanov
    Sep 11, 2013 at 10:00

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