I have seen on the French Wikipedia that Ehresmann's fibration theorem is stated with the assumption that everything is $C^2$, see Théorème de Ehresmann. (On the English Wikipedia, the assumption is smooth, which I suppose means $C^\infty$, see Ehresmann's Lemma.)

Here is a translation of the French Wikipedia:

Ehresmann's fibration theorem states that a $C^2$ map $f:M \to N$ where $M$ and $N$ are $C^2$ differential manifolds, and such that $f$ is a surjective submersion and $f$ is proper, is a locally trivial fibration.

(what is meant is that it "is a locally trivial $C^2$ fibration", I just checked Ehresmann's statement in his article Les connexions infinitésimales dans an espace fibré différentiable, Seminaire N. Bourbaki, 1948-1951, exp. n° 24, p. 153-168.)

  1. Does anybody know of a counterexample in the case where the smoothness is only $C^1$? I mean a $C^1$ map as above which would not be $C^1$ fibration.

  2. I am specially interested in the case where the domain of the submersion has dimension 2 and the range dimension 1. I suspect that the theorem holds in this case (select one fiber and build a local fibration-trivialization around it by patching the x-coordinate of local submersion-trivializations where level curves would be horizontals in $\mathbb{R}^2$). Is that already proved or disproved somewhere?

  • 2
    $\begingroup$ Could you state a precise version of Ehresmann's theorem, for the sake of making your question concrete? Ehresmann's theorem, regardless of which version you use is basically just an application of the implicit function theorem. You can use the tubular neighbourhood theorem to speed up the proof and make it a little more conceptual. If you add in Riemann metrics and use the exponential map you can (very much!) speed up the proof but you bring in the $C^2$ assumption. $\endgroup$ Oct 14 '14 at 6:05
  • $\begingroup$ Sure, I'll edit the question and translate the French Wikipedia statement here. $\endgroup$ Oct 14 '14 at 8:55

I believe the paper "Foliations and Fibrations" by Earle and Eells, Jr. in J. Differential Geometry 1 (1967), pp. 33-41 can be helpful.

The first proposition in their fourth section is a very general extension of Ehresmann:

Theorem: If $f:X \to Y$ is a proper $C^1$-map of Finsler manifolds which foliates $X$, then $f$ is a locally trivial $C^0$-fibration.

Here Earles and Eells say $f$ foliates $X$ if (i) for every $x\in X$ the differential $df_x$ maps the tangent space $T_x X$ surjectively onto $T_{fx}Y$ and (ii) the fibres $\{f^{-1}(y)\}_{y\in Y}$ are closed differentiable submanifolds of $X$ defining a foliation whose leaves are the connected components of the manifolds $f^{-1}(y)$.

They remark (bottom of pp. 38) that there are theorems asserting $C^k$ maps $f$ foliating $X$ define locally trivial $C^k$-foliations provided that one can find $C^k$-partitions of unity. They refer to R. Hermann, "A sufficient condition that a mapping of riemannian manifolds be a fibre bundle", Proc. Amer. Math. Soc. 11 (1960) pp. 236-242.

So while their main theorem is very general (defined for Finsler manifolds modelled on Banach spaces with continuously varying family of norms on the tangent spaces) , there seems hope that in concrete finite dimensional settings one has $C^1$ foliatings maps defining $C^1$-locally trivial fibrations.

  • $\begingroup$ Thanks for the very precise references. The paper by Hermann takes place in the $C^\infty$ class (and makes strong assumptions on the foliation: compatibility with the inner product). The remark by Earle and Eells is more encouraging. They seem to repeat it on page 39. $\endgroup$ Oct 14 '14 at 8:53

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