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

2. Since manifolds are locally diffeomorphic to $R^n$ there can always locally foliated. But having a submersion imposes global topologic restriction. How can you define a submersion from $S^2$ to $R^1$?

3. Hello,

thank you for your answer. I don't know how to define a submersion from $S^2$ to $\mathbb{R}$.

When i was thinking about submersions and foliations i had the simple example of a open cube in mind.

For instance if we have the manifold $(0,1)^n \subset \mathbb{R}^n$

Then we can define various submersions like:

1) $f0,1)^n->(0,1)^n, f=id" alt="f0,1)^n->(0,1)^n, f=id" />
2) $f0,1)^n->\mathbb{R},\; f(x_1,....,x_n):=x_1\; or\;
3)f0,1)^n->\mathbb{R}^2\; f(x_1,...,x_n):=(x_1,x_2)" alt="f0,1)^n->\mathbb{R},\; f(x_1,....,x_n):=x_1\; or\;
3)f0,1)^n->\mathbb{R}^2\; f(x_1,...,x_n):=(x_1,x_2)" /> and so on.

This maps are submersions between manifolds , whereas the image of f has different dimension.

Ok, this examples are really simple. I don't know whether one can define "various" submersions, if we have complicated manifolds. (even if they are so descriptive like your example $S^2$
I have not so much experience with such situations.

If i have understand your comment correct, it is in general difficult to define submersions, right?

Regards

4. Since $(0,1)^n$ is diffeomorphic to $R^n$, so your examples are still the simple (or even trivial) cases.
As I said, having submersion introduces some topological restriction. That is, not all of the manifolds have non-trivial submersions.
Take $M=S^2, N=R^1$ as a simple example. Since M is compact any real function f of M must have maxmimal and minimal values. Thus f must have critical points. This makes f fail to be a submersion.

5. Hello,

To consider again our example $M=S^2$. The image of M under any real function f has to be compact again, i.e. $f(S^2)=[a,b]$
If our maximal (minimal) value is a real maximum (minimum) then we have critical values, as you said.
But what happens if our maximal and minimal values are a and b?
Then I think the derivative doesn't vanish anymore!?. Why is it impossoble to have such a case?

Regards

6. Try to prove that "the derivative doesn't vanish anymore".

7. Thank you very much! It must vanish of course. I had the wrong picture in mind.

Regards