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If $f \colon M \to N$ is a smooth map between smooth manifolds, is it possible to find an open and dense set $M_0$, such that $f(M_0)$ is a manifold and $f \colon M_0 \to f(M_0)$ is a surjective submersion?

Edit: Since I think it isn't possible to always find such an open set, what assumptions do we need, to find such an open set?

More specifically, I have an hamiltonian Lie group action of $G$ on $M$ with $G$-equivariant momentum map $\Phi \colon M \to \mathfrak{g}^*$. In my paper the authors are assuming that $\Phi(M) = W$ is a manifold and $\Phi$ is a submersion on the image. But by the properties of the momentum map that means, that $\dim W = \dim G.x \quad \forall x \in M$. Since that means, that the $G$-orbits in $M$ should have all the same dimension, I think I have to restrict myself to the subset $M_0$ of maximal dimensional orbits. Then I would need that $f(M_0)$ is first a submanifold and second of dimension equal to the maximaldimensional $G$-orbits in $M$.

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  • $\begingroup$ For surfaces $M, N$, if the rank of $M \to N$ is 1 on some open set, with image in a curve, and the rank is 2 somewhere else, then there is no $M_0$. $\endgroup$ – Ben McKay Apr 29 '16 at 9:17
  • $\begingroup$ Being a submersion is a local condition. So you want to know a) under which conditions $f$ is a submersion at a point $x \in M$ and b) under which conditions the set of such points is dense in $M$. I guess that this question is a bit broad. Maybe you are in a more specific situation? $\endgroup$ – Helene Sigloch Apr 29 '16 at 10:18
  • $\begingroup$ @HeleneSigloch: I just edited the more specific setting. $\endgroup$ – Olorin May 1 '16 at 13:55
  • $\begingroup$ If $f$ is generic, then such an $M_0$ exists. If I remember correctly, $M_0$ then is the top stratum of a hierarchy of Thom-Boardman singularities. On the other hand, moment maps are typically not generic as far as I know. $\endgroup$ – Sebastian Goette May 1 '16 at 18:53

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