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

In summary, constructing a function on (smooth) manifolds that is a submersion without being (the projection map of) a fiber bundle can be done by taking any old map and removing the set of critical points, or by using any smooth diffeomorphism or inclusion map. There are also countless other ways to achieve this.
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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.
 
  • #3
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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1. What is submersion?

Submersion is a type of smooth map between two smooth manifolds. It is a mapping that preserves differentiable structure, and has a surjective derivative. In simpler terms, it is a continuous and smooth function that "covers" every point in the target space.

2. How is submersion related to fiber bundles?

Submersion and fiber bundles are closely related concepts. In a fiber bundle, the total space is made up of a collection of "copies" of the base space, each attached to a point in the base space. Submersion is used to define the base space and the way in which the copies of the base space are attached, making it an essential component of understanding fiber bundles.

3. What is the purpose of studying submersion and fiber bundles?

Submersion and fiber bundles are important concepts in differential geometry and topology. They provide a framework for understanding the behavior of smooth maps and spaces, and are crucial in many areas of mathematics and physics, such as in the study of manifolds and vector bundles.

4. Can you give an example of a submersion?

A common example of a submersion is a projection map from a sphere to a plane. The sphere can be thought of as the total space, and the plane as the base space. Each point on the sphere is "mapped" to a point on the plane, covering the entire plane and satisfying the conditions of a submersion.

5. Are there any applications of submersion and fiber bundles in real life?

Yes, there are many real-life applications of submersion and fiber bundles. One example is in the study of fluid dynamics, where submersions are used to model the flow of water and other fluids. Additionally, fiber bundles are used in engineering and physics, such as in the study of electromagnetism and quantum mechanics.

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