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 $C^1$?

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?

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?

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

Théorème de Ehresmann

Ehresmann's Lemma

1. Does anybody know of a counterexample in the case where the smoothness is only $C^1$?

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?

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 $C^1$?

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?