Fundamental Theoreom on Flows

The Fundamental Theorem on Flows

Every smooth vector field on a smooth manifold has a unique smooth maximal flow with some nice properties.

Formally

Let be a smooth vector field on a smooth manifold . There exists a unique flow whose infinitesimal generator is . This flow has the properties that

1. is the unique maximal integral curve of with
2. If , then is the interval . In this way, “following the integral curve” by a value of shifts the domain of where the new integral curve is defined on by .
3. For each , the set of all points in that where an integral curve is defined for is open.
4. The function is a diffeomorphism with inverse .

Proof

todo It’s HARD…

References

@lee2013 - Chapter 9