Submersion between manifolds
I have a question about submersions. A submersion f:M->N between manifolds defines a foliation on M of codimension dimN.
I agree with this. But in my proof i showed that the origin atlas on M is already a foliation atlas (by using that we have a submersion of course!).
Now i'm not sure about the implications of this proposition. I think i can define a lot of submersions f:M->N' with , right? Therefore the given atlas on M can be considered as foliation atlas of codimension (for different submersions with different dimension of the range we get different codimension)
I.e. i can consider one and the same atlas as a foliation atlas of various codimensions, if i find the corresponding submersion.
This sounds a little bit odd to me.
Therefore my question is, whether my thoughts are right? Can you interpret this in a reasonable way?