Vector Fields!

Lee Smooth Manifolds

Geometry feels a lot easier after getting humbled by analysis.

(Lee 9.38) If $M$ is a smooth manifold and $V,W\in \mathfrak{X}(M)$, then ${\mathcal{L}}_{V}W=[V,W]$.

Proof) Recall that if $U$ is a smooth chart and $p\in U$, in the coordinates of $U$, for any $f\in C^{\infty }(M)$

$$[V,W{]}_p=V_{p}W-W_{p}V=V^i(p){\frac{\partial }{\partial x^i}|}_p\left(W^j\frac{\partial }{\partial x^j}\right)-W^i(p){\frac{\partial }{\partial x^i}|}_p\left(V^j\frac{\partial }{\partial x^j}\right)$$ $$=V^i(p)\frac{\partial W^j}{\partial x^i}(p){\frac{\partial }{\partial x^j}|}_p-W^i(p)\frac{\partial V^j}{\partial x^i}(p){\frac{\partial }{\partial x^j}|}_p$$

while

$$({\mathcal{L}}_{V}W{)}_p={\frac{d}{dt}|}_{t=0}({\theta }_{-t}{)}_{\theta (t,p)}{W}_{\theta (t,p)}$$

Let $\mathcal{R}(V)\subset M$ be the set of regular points of $V$ (where $V_p\ne 0$). If $p\in \mathcal{R}(V)$, take a smooth chart $p\in U$ such that $V=\partial /\partial x^1$, in which (with a mild abuse of notation)

$$({\mathcal{L}}_{V}W{)}_p={\frac{d}{dt}|}_{t=0}{W}_{p+t{e}^1}={\frac{\partial }{\partial x^1}|}_{p}W=[\frac{\partial }{\partial x^1},W{]}_p=[V,W{]}_p$$

If $p\in \text{supp}(\mathcal{R}(V))$, $({\mathcal{L}}_{V}W{)}_p=[V,W{]}_p$ by continuity, while if $p\notin \text{supp}(\mathcal{R}(V))$ there is a neighborhood of $p$ in which $V=0$; in this case $({\mathcal{L}}_{V}W{)}_p=0=[V,W{]}_p$.

(Lee 9.42) For smooth vector fields $V$ and $W$ on a smooth manifold $M$, $V$ and $W$ commute (i.e. $[V,W]=0$) iff $W$ is invariant under the flow of $V$.

Proof) Let $\theta$ be the flow of $V$ and $W$ be invariant under the flow of $V$; for any $(t,p)\in \mathcal{D}$:

$$(d{\theta }_t{)}_p(W_p)=W_{{\theta }_t(p)}\Rightarrow (d{\theta }_{-t}{)}_{{\theta }_t(p)}(W_{{\theta }_t(p)})=W_p\Rightarrow ({\mathcal{L}}_{V}W{)}_p=0$$

Thus, $V$ and $W$ commute. Conversely, assume ${\mathcal{L}}_{V}W=0$ and consider some $p\in M$. For any $t_0\in {\mathcal{D}}^{(p)}$, note that

$${\frac{d}{dt}|}_{t=t_0}(d{\theta }_{-t}{)}_{{\theta }_t(p)}(W_{{\theta }_t(p)})=(d{\theta }_{-t_0})\circ {\frac{d}{ds}|}_{s=0}(d{\theta }_{-s}{)}_{{\theta }_s({\theta }_{t_0}(p))}(W_{{\theta }_s({\theta }_{t_0}(p)}))=(d{\theta }_{-t_0})({\mathcal{L}}_{V}W{)}_{{\theta }_{t_0}(p)}$$

Thus, for any $t_0\in {\mathcal{D}}^{(p)}$,

$${\frac{d}{dt}|}_{t=t_0}(d{\theta }_{-t}{)}_{{\theta }_t(p)}(W_{{\theta }_t(p)})=0\Rightarrow (d{\theta }_{-t_0}{)}_{{\theta }_{t_0}(p)}(W_{{\theta }_{t_0}(p)})=W_p\Rightarrow W_{{\theta }_{t_0}(p)}=(d{\theta }_{t_0}{)}_p{W}_p$$

and we are done.

(Lee 9.44) Smooth vector fields commute iff their flows commute.

Proof) Let $M$ be a smooth manifold and $V,W\in \mathfrak{X}(M)$ with $\theta$ and $\psi$ being their respective flows. First, assume $V$ and $W$ commute. Let $J$ and $K$ be open intervals containing zero such that ${\theta }_t\circ {\psi }_s(p)$ is defined for all $(t,s)\in J\times K$. For any $t\in J$, define $\gamma :K\to M$ by $\gamma (s)={\theta }_t\circ {\psi }_s(p)$. Then, $\gamma (0)={\theta }_t(p)$ and

$${\gamma }^{\prime }(s)=(d{\theta }_t{)}_{{\psi }_s(p)}(W_{{\psi }_s(p)})=W_{{\theta }_t\circ {\psi }_s(p)}=W_{\gamma (s)}$$

as $W$ is invariant under the flow of $V$. $\gamma$ is therefore the unique integral curve starting at ${\theta }_t(p)$, and thus for any $(s,t)\in K\times J$, ${\psi }_s\circ {\theta }_t(p)$ is defined and ${\psi }_s\circ {\theta }_t(p)=\gamma (s)={\theta }_t\circ {\psi }_s(p)$. I.e. the flows commute.

Now assume $\theta$ and $\psi$ commute. For any $p\in M$, there exists open intervals $J$ and $K$ containing zero such that ${\theta }_t\circ {\psi }_s(p)$ is defined for all $(t,s)\in J\times K$. Then, for any $t\in J$

$$(d{\theta }_{-t}{)}_{{\theta }_t(p)}(W_{{\theta }_t(p)})={\frac{d}{ds}|}_{s=0}{\theta }_{-t}\circ {\psi }^{{\theta }_t(p)}(s)={\frac{d}{ds}|}_{s=0}{\psi }^p(s)=W_p\Rightarrow {\mathcal{L}}_V{W}_p=0$$

We are done.

(Lee 9.46) Let $M$ be a smooth $n$-manifold, and $(V_1,V_2,\dots ,V_k)$ be a linearly independent $k$-tuple of smooth commuting vector fields on an open subset $W\subset M$. For each $p\in W$, there exists a smooth coordinate chart $(U,(s^i))\subset W$ centered at $p$ such that $V_i=\partial /\partial s^i$ for $1\le i\le k$. If $S\subset W$ is a codimension-$k$ embedded submanifold and $p$ is a point of $S$ such that $T_{p}S$ is complementary to the span of $(V_1{|}_p,V_2{|}_p,\dots ,V_k{|}_p)$, then the coordinates can be chosen such that $S\cap U$ is the slice determined by $s^1=\dots =s^k=0$.

Proof) If such a submanifold doesn’t exist it’s not too difficult to make one ourselves; we assume we are given such an $S$. We begin with a slice chart $p\in (U,(x^i))\subset W$ centered at $p$ such that $U\cap S$ is the slice $x^1=\dots =x^k=0$. By assumption, $\{V_1{|}_p,\dots ,V_k{|}_p,\partial /\partial x^{k+1},\dots ,\partial /\partial x^n\}$ spans $T_{p}M$. Let ${\theta }_i$ be the flow of $V_i$; there exists $ϵ>0$ and a neighborhood $Y$ of $p$ in $U$ such that the composition $({\theta }_1{)}_{t_1}\circ \dots \circ ({\theta }_k{)}_{t_k}$ is defined on $Y$ and maps into $U$ whenever $|t_i|<ϵ$ for all $1\le i\le k$.

Define $\Omega \subset {\mathbb{R}}^{n-k}$ such that $\Omega :\{(s^{k+1},\dots ,s^n):(0,\dots ,s^{k+1},\dots ,s^n)\in Y\}$ and $\Phi :(-ϵ,ϵ{)}^k\times \Omega \to U$ such that $\Phi (s_1,\dots ,s_n)=({\theta }_1{)}_{s_1}\circ ({\theta }_2{)}_{s_2}\circ \dots \circ ({\theta }_k{)}_{s_k}(0,\dots ,s^{k+1},\dots ,s^n)$. We can see that for any $1\le i\le k$ and $q\in (-ϵ,ϵ{)}^k\times \Omega$:

$$d\Phi \left(\frac{\partial }{\partial s^i}{|}_q\right)f={\frac{\partial }{\partial s^i}|}_{q}f(\Phi (q))={\frac{\partial }{\partial s^i}|}_{q}f(({\theta }_i{)}_{s^i}\circ \dots (0,\dots ,s^{k+1},\dots ,s^n))=V_i{|}_{\Phi (q)}f$$

It suffices to show $\Phi$ is a diffeomorphism with some neighborhood of $p$ in $U$, which is equivalent to $d{\Phi }_p$ being invertible. This follows from the fact that $\{V_1{|}_p,\dots ,V_k{|}_p,\partial /\partial x^{k+1},\dots ,\partial /\partial x^n\}$ spans $T_{p}M$.