Building Charts from Vector Fields
Sorry for the delay; I’ve been swamped at my actual job the last couple days.
Anyway, I left off last time by pointing out that coordinate vector fields commute:
![\displaystyle\left[\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right]=0](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cleft%5B%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x%5Ei%7D%2C%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x%5Ej%7D%5Cright%5D%3D0&bg=e6e6e6&fg=333333&s=0 "\displaystyle\left[\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right]=0")
Today I want to show a certain converse: if we have 
vector fields 
on some open region 
that are linearly independent at some 
and which commute — ![[X_i,X_j]=0](https://s0.wp.com/latex.php?latex=%5BX_i%2CX_j%5D%3D0&bg=e6e6e6&fg=333333&s=0 "[X_i,X_j]=0")
for all 
and 
— then we can find some coordinate chart 
around 
so that the 
are the first coordinate vector fields. That is,
In particular, if 
is a vector field with 
then there is a coordinate chart around with 
.
If is any chart, then we can describe the th coordinate vector field by saying it’s the unique vector field on 
that is 
-related to the th partial derivative in 
. That is, we’re trying to prove that: 
on 
.
In fact, we can further simplify our claim by assuming that 
, 
, and 
— that the vector fields agree at the point 
. Indeed, if 
is any coordinate map taking to 
then we can define the vector fields 
on . These must have vanishing brackets because we can calculate:
![\displaystyle\begin{aligned}{}[Y_i,Y_j]&=[z_*\circ X_i\circ z^{-1},z_*\circ X_j\circ z^{-1}]\\&=z_*\circ[X_i,X_j]\circ z^{-1}\\&=z_*\circ0\circ z^{-1}=0\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%7B%7D%5BY_i%2CY_j%5D%26%3D%5Bz_%2A%5Ccirc+X_i%5Ccirc+z%5E%7B-1%7D%2Cz_%2A%5Ccirc+X_j%5Ccirc+z%5E%7B-1%7D%5D%5C%5C%26%3Dz_%2A%5Ccirc%5BX_i%2CX_j%5D%5Ccirc+z%5E%7B-1%7D%5C%5C%26%3Dz_%2A%5Ccirc0%5Ccirc+z%5E%7B-1%7D%3D0%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}{}[Y_i,Y_j]&=[z_*\circ X_i\circ z^{-1},z_*\circ X_j\circ z^{-1}]\\&=z_*\circ[X_i,X_j]\circ z^{-1}\\&=z_*\circ0\circ z^{-1}=0\end{aligned}")
What’s more, if 
is a local diffeomorphism of with 
, then 
is a coordinate map satisfying our assertion.
Now, let 
be the flow of , and let 
be a small enough neighborhood of that we can define 
by
](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28a_1%2C%5Cdots%2Ca_n%29%3D%5Cleft%5B%5CPhi%5E1_%7Ba_1%7D%5Ccirc%5Cdots%5Ccirc%5CPhi%5Ek_%7Ba_k%7D%5Cright%5D%280%2C%5Cdots%2C0%2Ca_%7Bk%2B1%7D%2C%5Cdots%2Ca_n%29&bg=e6e6e6&fg=333333&s=0 "\displaystyle f(a_1,\dots,a_n)=\left[\Phi^1_{a_1}\circ\dots\circ\Phi^k_{a_k}\right](0,\dots,0,a_{k+1},\dots,a_n)")
The order that the flows come in here doesn’t matter, since we’re assuming that the — and thus their flows — commute. Anyway, given any smooth test function 
on we can check
-\left[\phi\circ f\right](a_1,a_2,\dots,a_n)\right]\\&=\lim\limits_{h\to0}\frac{1}{h}\left[\left[\phi\circ\Phi^1_{a_1+h}\circ\dots\circ\Phi^k_{a_k}\right](0,\dots,0,a_{k+1},\dots,a_n)-\left[\phi\circ f\right](a)\right]\\&=\lim\limits_{h\to0}\frac{1}{h}\left[\left[\phi\circ\Phi_h\circ\Phi^1_{a_1}\circ\dots\circ\Phi^k_{a_k}\right](0,\dots,0,a_{k+1},\dots,a_n)-\left[\phi\circ f\right](a)\right]\\&=\lim\limits_{h\to0}\frac{1}{h}\left[\left[\phi\circ\Phi_h\right](a)-\left[\phi\circ f\right](a)\right]\\&=\left(X_1\right)_{f(a)}(\phi)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%28f_%2AD_1%5Cright%29_a%5Cphi%26%3D%5Cleft%28D_1%5Cright%29_a%28%5Cphi%5Ccirc+f%29%5C%5C%26%3D%5Clim%5Climits_%7Bh%5Cto0%7D%5Cfrac%7B1%7D%7Bh%7D%5Cleft%5B%5Cleft%5B%5Cphi%5Ccirc+f%5Cright%5D%28a_1%2Bh%2Ca_2%2C%5Cdots%2Ca_n%29-%5Cleft%5B%5Cphi%5Ccirc+f%5Cright%5D%28a_1%2Ca_2%2C%5Cdots%2Ca_n%29%5Cright%5D%5C%5C%26%3D%5Clim%5Climits_%7Bh%5Cto0%7D%5Cfrac%7B1%7D%7Bh%7D%5Cleft%5B%5Cleft%5B%5Cphi%5Ccirc%5CPhi%5E1_%7Ba_1%2Bh%7D%5Ccirc%5Cdots%5Ccirc%5CPhi%5Ek_%7Ba_k%7D%5Cright%5D%280%2C%5Cdots%2C0%2Ca_%7Bk%2B1%7D%2C%5Cdots%2Ca_n%29-%5Cleft%5B%5Cphi%5Ccirc+f%5Cright%5D%28a%29%5Cright%5D%5C%5C%26%3D%5Clim%5Climits_%7Bh%5Cto0%7D%5Cfrac%7B1%7D%7Bh%7D%5Cleft%5B%5Cleft%5B%5Cphi%5Ccirc%5CPhi_h%5Ccirc%5CPhi%5E1_%7Ba_1%7D%5Ccirc%5Cdots%5Ccirc%5CPhi%5Ek_%7Ba_k%7D%5Cright%5D%280%2C%5Cdots%2C0%2Ca_%7Bk%2B1%7D%2C%5Cdots%2Ca_n%29-%5Cleft%5B%5Cphi%5Ccirc+f%5Cright%5D%28a%29%5Cright%5D%5C%5C%26%3D%5Clim%5Climits_%7Bh%5Cto0%7D%5Cfrac%7B1%7D%7Bh%7D%5Cleft%5B%5Cleft%5B%5Cphi%5Ccirc%5CPhi_h%5Cright%5D%28a%29-%5Cleft%5B%5Cphi%5Ccirc+f%5Cright%5D%28a%29%5Cright%5D%5C%5C%26%3D%5Cleft%28X_1%5Cright%29_%7Bf%28a%29%7D%28%5Cphi%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}\left(f_*D_1\right)_a\phi&=\left(D_1\right)_a(\phi\circ f)\\&=\lim\limits_{h\to0}\frac{1}{h}\left[\left[\phi\circ f\right](a_1+h,a_2,\dots,a_n)-\left[\phi\circ f\right](a_1,a_2,\dots,a_n)\right]\\&=\lim\limits_{h\to0}\frac{1}{h}\left[\left[\phi\circ\Phi^1_{a_1+h}\circ\dots\circ\Phi^k_{a_k}\right](0,\dots,0,a_{k+1},\dots,a_n)-\left[\phi\circ f\right](a)\right]\\&=\lim\limits_{h\to0}\frac{1}{h}\left[\left[\phi\circ\Phi_h\circ\Phi^1_{a_1}\circ\dots\circ\Phi^k_{a_k}\right](0,\dots,0,a_{k+1},\dots,a_n)-\left[\phi\circ f\right](a)\right]\\&=\lim\limits_{h\to0}\frac{1}{h}\left[\left[\phi\circ\Phi_h\right](a)-\left[\phi\circ f\right](a)\right]\\&=\left(X_1\right)_{f(a)}(\phi)\end{aligned}")
That is, 
. To see this for any other , simply swap around the flows to bring to the front.
We can also check that
-\left[\phi\circ f\right](0,\dots,0)\right]\\&=\lim\limits_{h\to0}\frac{1}{h}\left[\phi(0,\dots,0,h,0,\dots,0)-\phi(0,\dots,0)\right]\\&=\left(D_{k+i}\right)_0\phi\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%28f_%2AD_%7Bk%2Bi%7D%5Cright%29_0%5Cphi%26%3D%5Cleft%28D_%7Bk%2Bi%7D%5Cright%29_0%28%5Cphi%5Ccirc+f%29%5C%5C%26%3D%5Clim%5Climits_%7Bh%5Cto0%7D%5Cfrac%7B1%7D%7Bh%7D%5Cleft%5B%5Cleft%5B%5Cphi%5Ccirc+f%5Cright%5D%280%2C%5Cdots%2C0%2Ch%2C0%2C%5Cdots%2C0%29-%5Cleft%5B%5Cphi%5Ccirc+f%5Cright%5D%280%2C%5Cdots%2C0%29%5Cright%5D%5C%5C%26%3D%5Clim%5Climits_%7Bh%5Cto0%7D%5Cfrac%7B1%7D%7Bh%7D%5Cleft%5B%5Cphi%280%2C%5Cdots%2C0%2Ch%2C0%2C%5Cdots%2C0%29-%5Cphi%280%2C%5Cdots%2C0%29%5Cright%5D%5C%5C%26%3D%5Cleft%28D_%7Bk%2Bi%7D%5Cright%29_0%5Cphi%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}\left(f_*D_{k+i}\right)_0\phi&=\left(D_{k+i}\right)_0(\phi\circ f)\\&=\lim\limits_{h\to0}\frac{1}{h}\left[\left[\phi\circ f\right](0,\dots,0,h,0,\dots,0)-\left[\phi\circ f\right](0,\dots,0)\right]\\&=\lim\limits_{h\to0}\frac{1}{h}\left[\phi(0,\dots,0,h,0,\dots,0)-\phi(0,\dots,0)\right]\\&=\left(D_{k+i}\right)_0\phi\end{aligned}")
Thus 
is the identity transformation on 
. The inverse function theorem now tells us that there is a chart around with 
, which will then satisfy our assertions.
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June 22, 2011 -
Posted by John Armstrong |
Differential Topology, Topology
[…] we can find a patch and an everywhere-nonzero vector field on so that spans for every . Then we know we can find a chart around such that on . Then the curve with coordinates and for all other […]
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