Flowout Theorem Part I , John Lee's smooth manifold, page 217.

Question

I am having trouble understanding John Lee's proof of Flowout theorem.

Theorem 9.20: Suppose $M$ is a smooth manifold $S \subseteq M$ is an embedded $k$-dimensional submanifold and $V \in \mathfrak{X}(M)$ is a smooth vector field that is nowhere tangent to $S$. Let $\theta: \mathfrak{D} \rightarrow M$ be the flow of $V$, let $\mathfrak{O} = (\Bbb R \times S) \cap \mathfrak{D}. $ Let $\Phi = \theta|_{\mathfrak O}$.

1. $\Phi: \mathfrak{O} \rightarrow M$ is an immersion.
2. $\frac{\partial}{\partial t } \in \mathfrak{X}(\mathfrak{O})$ is $\Phi$-related to $V$.

Why is $\mathfrak{O}$ a manifold - what manifold structure are we giving it? Why is the map $\Phi$ smooth being a restriction?

The latter question requires knowing a coordinate chart on $\mathfrak{O}$...

Answer 1

By Fundamental Theorem of Flows we know that $\theta : \mathfrak{D} \to M$ is a smooth map, with $\mathfrak{D}$ is an open subset of $\Bbb{R}\times M$, called the Flow Domain. Now $\mathfrak{D}$ is an open submanifold of $\Bbb{R}\times M$ and so to deal with $\Phi$, we look at $\mathfrak{O} = (\Bbb{R}\times S)\cap \mathfrak{D}$ as an embedded submanifold of $\mathfrak{D}$. We can see this by finding slice charts for $\mathfrak{O} \subseteq \mathfrak{D}$. The passage below is the sketch. Roughly, since $\Bbb{R} \times S$ is an embedded submanifold of $\Bbb{R}\times M$, its restriction $\mathfrak{O} = (\Bbb{R}\times S)\cap \mathfrak{D}$ to an open submanifold $\mathfrak{D} \subseteq \Bbb{R}\times M$ is also embedded submanifold of $\mathfrak{D}$, by local slice criterion.

Since $\mathfrak{D} \subseteq \Bbb{R} \times M$ is open, $\mathfrak{D}$ is an open submanifold of $\Bbb{R} \times M$. Now $\mathfrak{O} = (\Bbb{R} \times S) \cap \mathfrak{D}$ is an embedded submanifold of $\mathfrak{}$ since it is satisfy local slice condition. Suppose you have a point $(t,p) \in \mathfrak{O} \subseteq \Bbb{R} \times S$. Since $\Bbb{R} \times S$ is an embedded submanifold of $\Bbb{R} \times M$, we have a smooth chart $(U,\varphi)$ in $M$ contain $p$ such that $(\Bbb{R} \times (U\cap S))$ is a single $(k+1)$-slice in $\Bbb{R} \times U$. Now the intersection $(\Bbb{R} \times U)\cap \mathfrak{D}$ is a slice chart for $\mathfrak{O}$ in $\mathfrak{D}$.

The map $\Phi = \theta|_{\mathfrak{O}}$ is smooth, since $(1)$ The flow $\theta : \mathfrak{D} \to M$ is a smooth map by Fundamental Theorem of Flows. $(2)$ $\mathfrak{O}$ is an embedded submanifold of $\mathfrak{D}$, and the restriction of any smooth map to an embedded submanifold is also smooth.