When is a foliated manifold a fibre bundle?

In summary: Mobius band is not a circle bundle, but Ehresmann's lemma tells us that the Mobius band is a circle bundle over the interval. So the foliation must provide some additional information that is not captured by the Ehresmann's lemma.
  • #1
center o bass
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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?
 
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  • #2
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?
 
  • #3
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?
 
  • #4
Ah, ok I get where you're going. Sadly, I do not know the answer to this question...sorry.
 
  • #5
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.
 
  • #6
See, simple foliation and Ehresmann's lemma
 

1. What is a foliated manifold?

A foliated manifold is a mathematical space that is locally modeled on a product space, where the factors of the product are a base space and a fiber space. The base space is often a lower-dimensional space, such as a curve or surface, and the fiber space is often a higher-dimensional space, such as a higher-dimensional manifold.

2. What is a fibre bundle?

A fibre bundle is a mathematical space that is locally modeled on a product space, where the factors of the product are a base space and a fiber space. The base space is often a lower-dimensional space, such as a curve or surface, and the fiber space is often a higher-dimensional space, such as a higher-dimensional manifold. The difference between a foliated manifold and a fibre bundle is that in a foliated manifold, the base space is the same for each local model, while in a fibre bundle, the base space can vary.

3. When is a foliated manifold a fibre bundle?

A foliated manifold is a fibre bundle when the base space is the same for each local model. In other words, the leaves of the foliation must be the same for each local model. This means that the foliation must be "smooth" and "consistent" throughout the manifold. This condition is known as the "transversality condition."

4. What is the importance of understanding when a foliated manifold is a fibre bundle?

Understanding when a foliated manifold is a fibre bundle is important in many areas of mathematics, including topology, geometry, and differential equations. This knowledge can be used to prove theorems and make mathematical constructions, and it also has applications in physics and engineering, where fibre bundles are used to describe physical phenomena and models.

5. Are there any practical applications of foliated manifolds being fibre bundles?

Yes, there are many practical applications of foliated manifolds being fibre bundles. For example, in physics, fibre bundles are used to describe the behavior of fields, such as electromagnetic fields and gravitational fields. In engineering, fibre bundles are used to model complex systems, such as fluid flow and heat transfer. In computer science, fibre bundles are used to represent and manipulate data structures. In all of these applications, understanding when a foliated manifold is a fibre bundle is crucial for making accurate predictions and developing efficient solutions.

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