Talk:Ehresmann's lemma

Can you get this result by just apply Reeb stability theorem?

Here is a proof sketch. Can someone verify (I'm just a student, though I am following an exercise sheet) and then add to the page?

It can be assumed that $N = R^n$. Let $X_i$ denote the coordinate vector fields on $R^n$. Using the local normal form for submersions, plus a partition of unity argument, one can lift the $X_i$ to vector fields on $M$, $\tilde{X}_i$. Then properness of the fibers implies that flows exist for all time for the $\tilde{X}_i$. Call the diffeomorphism obtained by flowing for time t along $\tilde{X}_i$ $\phi_i(t)$. Fix a point $q \in R^n$, and denote by $F$ the fiber. Then for any point $p \in R^n$, denote by $x_i(p)$ its coordinates. Then $\phi_1(x_1(p)) \circ \ldots \circ \phi_n(x_n(p))$ sends $F$ to $f^{-1}(p)$ diffeomorphically (the inverse is obtained by flowing back). This defines a diffeomorphism $M x F \to M$ over $R^n$. — Preceding unsigned comment added by AreaMathMan (talk • contribs) 15:04, 27 November 2016 (UTC)


 * I'm no mathematician, but that's not that method of adding material to wikipedia. You need to find a reliable source, such a proof from a text book, that you can reference. There is a wikipedia policy of No original research. Klbrain (talk) 10:39, 17 September 2017 (UTC)