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 $X$ be a smooth vector field on a smooth manifold $M$. There exists a unique flow ${\Phi }_X:\mathcal{D}\to M$ whose infinitesimal generator is $X$. This flow has the properties that

1. ${\Phi }_X^{(p)}$ is the unique maximal integral curve of $X$ with ${\Phi }_X^{(p)}(0)=p$
2. If $s\in {\mathcal{D}}^{(p)}$, then ${\mathcal{D}}^{({\Phi }_X(s,p))}$ is the interval ${\mathcal{D}}^{(p)}-s=\{t-s|t\in {\mathcal{D}}^{(p)}\}$. In this way, “following the integral curve” by a value of $s$ shifts the domain of where the new integral curve is defined on by $s$.
3. For each $t\in \mathbb{R}$, the set $M_t=\{p\in M|(t,p)\in \mathcal{D}\}$ of all points in $M$ that where an integral curve is defined for $t$ is open.
4. The function ${\Phi }_t:M_t\to M_{-t}$ is a diffeomorphism with inverse ${\Phi }_{-t}$ .

Proof

todo It’s HARD…

References

@lee2013 - Chapter 9