Foliation, fibration, fiber bundle?

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SUMMARY

The discussion clarifies the distinctions between foliation, fibration, and fiber bundles in the context of manifold theory. A foliation divides a manifold into topologically different equivalence classes known as leaves, while a fibration requires these classes to be topologically identical, thus forming a fiber bundle. An example provided illustrates that in a plane, the sets defined by constant distance from a point represent a foliation, whereas excluding the point transforms it into a fibration.

PREREQUISITES
  • Understanding of manifold theory
  • Familiarity with topological equivalence classes
  • Knowledge of fiber bundles
  • Basic concepts of foliation and fibration
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  • Study the properties of fiber bundles in differential geometry
  • Explore examples of foliations in various manifolds
  • Learn about the applications of fibrations in algebraic topology
  • Investigate the relationship between foliations and dynamical systems
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Mathematicians, particularly those specializing in topology and differential geometry, as well as students seeking to deepen their understanding of manifold structures and their applications.

jojoo
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what's the difference among those three objects?
Any body can give me some examples?

Thanks
 
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Both a foliation and a fibration of a manifold are a division of the manifold into sets of equivalence classes. For a foliation the equivalence classes can be topological different but for fibration they must be the same. A fibration gives the original manifold the structure of a fiber bundle, with each equivalence class being a fiber. The equivalence classes of a foliation are called leaves.

For a simple example, let r be the distance to some point in the plane. Then the sets \{r=\text{const}\} are a foliation, with leaves being circles and a point. If the point is excluded from the plane - turning it into a punctered plane - this will be a fibration.
 

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