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December 31st, 2014, 12:08 AM   #1
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How to solve this problem?

Proving that $\Phi_{t*}[\mathbb{Y},\mathbb{Z}]=[\Phi_{t*}\mathbb{Y},\Phi_{t*}\mathbb{Z}]$

Let $\mathbb{X},\mathbb{Y}$ be vector fields on $\mathbb{R}^n$. Let $\Phi_t$ denote the flow of $\mathbb{X}$. Define $L_{\mathbb{X}}\mathbb{Y}=[\mathbb{X},\mathbb{Y}]$. You are given that

$\displaystyle L^j_{\mathbb{X}}[\mathbb{Y},\mathbb{Z}]=\sum_{k=0}^{j} \binom jk [L^k_\mathbb{X}\mathbb{Y},L^{j-k}_\mathbb{X}\mathbb{Z}]$

and that $\displaystyle \Phi_{t*}\mathbb{Y} = \sum_{j=0}^{\infty}\frac{(-t)^j}{j!}L^j_{\mathbb{X}}\mathbb{Y}$

Show that $\displaystyle \Phi_{t*}[\mathbb{Y},\mathbb{Z}]=[\Phi_{t*}\mathbb{Y},\Phi_{t*}\mathbb{Z}]$

Let $F$ be a diffeomorphism on $\mathbb{R}^n$, define
$$ \mathbb{X}^i(x)=\frac{\partial F^i}{\partial x^1}(F^{-1}(x)), \mathbb{Y}^i(x)=\frac{\partial F^i}{\partial x^2}(F^{-1}(x))$$
Show that $[\mathbb{X},\mathbb{Y}]=0$.

$\begin{align}
\displaystyle \Phi_{t*}[\mathbb{Y},\mathbb{Z}] &= \sum_{j=0}^{\infty}\frac{(-t)^j}{j!}L^j_{\mathbb{X}}[\mathbb{Y},\mathbb{Z}] \\
&= \sum_{j=0}^{\infty}\frac{(-t)^j}{j!} \left(\sum_{k=0}^{j} \binom jk [L^k_\mathbb{X}\mathbb{Y},L^{j-k}_\mathbb{X}\mathbb{Z}]\right) \\
&= \sum_{l,k=0}^{\infty} \frac{(-t)^{k+l}}{(l+k)!} \binom{l+k}{k}[L^k_\mathbb{X}\mathbb{Y},L^{l}_\mathbb{X}\mathbb{Z }] \\
&= \sum_{l,k=0}^{\infty} \frac{(-t)^{k}(-t)^{l}}{l!k!} [L^k_\mathbb{X}\mathbb{Y},L^{l}_\mathbb{X}\mathbb{Z }] \\
&= [\Phi_{t*}\mathbb{Y},\Phi_{t*}\mathbb{Z}]
\end{align}$

I cannot see how to proceed now. For the last part I assume somehow the first part has been used as I just cannot see how
Ganesh Ujwal is offline  
 
December 31st, 2014, 07:17 PM   #2
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Looks absolutely awesome. Unfortunately, I'm not at your level.
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