Submersion between manifolds

Hello,

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 $\displaystyle dimN' \leq dimM$, right? Therefore the given atlas on M can be considered as foliation atlas of codimension $\displaystyle \leq dimM$ (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?

Regards