When is a foliated manifold a fibre bundle?

[center o bass]

I k-foliation of a $n$-manifold $M$ is a collection of disjoint, non-empty, submanifolds who's union is $M$, such that we can find a chart $(U,x^1, \ldots mx^k, y^{k+1}, \ldots, y^n)=(\phi, (x^\mu, y^\nu))$ about any point with the property that setting the $n-k$ last coordinates equal to a constant determines a particular submanifold, and varying $x^\mu$ let's us move around on the surface.

Now I wonder, what conditions would one additionally have to impose for $M$ to become a fibre bundle? Is it's enough that the leaves (submanifolds) of the foliation are diffeomorphic to each other?

 

[Matterwave]

I'm not sure what this construction has to do with fiber bundles. What is your base space? What is your fiber? What is your projection, and trivialization and structure group?

 

[center o bass]

If one has a foliation, then there excists a chart $(U,\phi)$ with the property that $\phi(U) = V \times W$, where the points in $W$ determines which leaf we are on. Now since these leaves are submanifold the chart defined by $(U\cap S, \tilde \phi)=( U\cap S, x^\mu)$ for a particular leaf S is a chart that takes $U\cap S$ to $\phi(U\cap S) = V$ where $U\cap S$ is an open subset of the leaf S. Hence, locally $M \sim ((\text{subset of} \ S) \times W$. Now if we impose conditions so that this is instead the product $S \times W$, and all the leafs is diffeomorphic to each other (in particular to $S$), we have a local trivialization with fibre S. If we then construct the quotient $B=M/S$ (the manifold of fibres), we have a base-space. We need no structure group for it to be a fibre bundle..

So the question is really what conditions are necessary to impose in order for us to have a product $S \times W$ instead of $\text{subset of} \ S) \times W$ and for the leaves to be diffeomorphic to each other. Does the one imply the other, or do we have to impose other coniditions?

 

[Matterwave]

Ah, ok I get where you're going. Sadly, I do not know the answer to this question...sorry.

 

[lavinia]

The leaves can be diffeomorphic without the foliation being a fiber bundle.

Example:

The Mobius band is foliated by circles but is not a circle bundle over a closed interval. This foliation extends to the Klein bottle where it is not a circle bundle over the circle.

 

[TamTamTam]

See, simple foliation and Ehresmann's lemma