Another quick definition: we say that a smooth map is a “submersion” if it is surjective, and if every point is a regular point of . Despite the similarity of the terms “immersion” and “submersion”, these are very different concepts, so be careful to keep them separate.
The nice thing about submersions is that every value of is a regular value, and every one has a nonempty preimage. Thus our extension of the implicit function theorem applies to show that is an -dimensional submanifold of .
One obvious example of submersion is a projection from a product manifold. As we’ve seen, the determinant of this projection is always a surjection. In fact, it’s a projection itself.
Another example is the projection from the tangent bundle down to its base manifold . Indeed, given any tangent vector at we can pick a coordinate patch around and the corresponding patch of . Within these coordinates we can easily calculate the derivative of and see that it’s just a projection onto the first components, which is surjective. In this case, the preimages are the stalks of the tangent bundle .