Since manifolds are locally diffeomorphic to there can always locally foliated. But having a submersion imposes global topologic restriction. How can you define a submersion from to ?
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 , 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?
Regards
Hello,
thank you for your answer. I don't know how to define a submersion from to .
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
Then we can define various submersions like:
1) 0,1)^n->(0,1)^n, f=id" alt="f0,1)^n->(0,1)^n, f=id" />
2) 0,1)^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
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
Since is diffeomorphic to , 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 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.
Hello,
thanks again for your explanations.
To consider again our example . The image of M under any real function f has to be compact again, i.e.
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