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

[Big-T]

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?

 

[zhentil]

The simplest way is to take any old map and remove the set of critical points.

 

[wofsy]

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.