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 
.
May 2, 2011
Posted by John Armstrong |
Differential Topology, Topology |
2 Comments
