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Foliation example

Joppy

Well-known member
MHB Math Helper
Mar 17, 2016
256
Are there any simple and possibly practical examples of a foliation? From the definitions here and here I can't help but think a foliation is a collection of sets which partition some space (a manifold) that satisfy certain conditions. What is their role and why are they important?
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,632
Leiden
Are there any simple and possibly practical examples of a foliation? From the definitions here and here I can't help but think a foliation is a collection of sets which partition some space (a manifold) that satisfy certain conditions. What is their role and why are they important?
Hey Joppy ,

I may be too quick to respond to your question.
After all, we can never tell when someone else may have an insight.
At this time I'm assuming that no one here really knows.

Anyway, let me ask: can you answer your own question?
What are foliations?
And what are their roles? (Wondering)
 

Joppy

Well-known member
MHB Math Helper
Mar 17, 2016
256
Hey Joppy ,

I may be too quick to respond to your question.
After all, we can never tell when someone else may have an insight.
At this time I'm assuming that no one here really knows.

Anyway, let me ask: can you answer your own question?
What are foliations?
And what are their roles? (Wondering)
Does it allow us to study manifolds in a systematic way? For instance, studying the leaves of a foliation can be simpler than studying the manifold itself (assuming the foliation exists)?

For a simple example, could we try define $\left\lbrace F_\alpha \right\rbrace$ for a sphere $S^3$?