Ehresmann's fibration theorem in the C1 class

Question

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?

Answer 1

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.