How Can a Function Be a Submersion on Manifolds Without Forming a Fiber Bundle?

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SUMMARY

This discussion focuses on constructing a function on smooth manifolds that acts as a submersion without being the projection map of a fiber bundle. It establishes that any smooth diffeomorphism qualifies as a submersion, highlighting the extensive list of possible submersions available. Additionally, the inclusion map can be utilized by excising a small ball around any point, which is effective for finitely many excised balls. The conversation emphasizes the multitude of methods available for achieving this goal.

PREREQUISITES
  • Smooth manifolds
  • Submersions and fiber bundles
  • Smooth diffeomorphisms
  • Inclusion maps
NEXT STEPS
  • Explore the properties of smooth diffeomorphisms in manifold theory
  • Study the construction and applications of inclusion maps in topology
  • Investigate critical points and their role in defining submersions
  • Learn about the relationship between submersions and fiber bundles in differential geometry
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Mathematicians, particularly those specializing in differential geometry, topology, and manifold theory, will benefit from this discussion.

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How would one go about to construct a function on (smooth) manifolds that is a submersion without being (the projection map of) a fiber bundle?
 
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The simplest way is to take any old map and remove the set of critical points.
 
there are a zillion ways.

E.G.

Any smooth diffeomorphism is a submersion. In any manifold there is a diffeomorphism that maps any point to any other. So the list of submersions is large.

If you don't want to use the whole manifold excise a small ball around any point and use the inclusion map . The process works for finitely many excised balls.

The are a zillion other ways as well.
 
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