# Lie Derivative of a Vector Field

• November 14th 2011, 06:43 AM
slevvio
Lie Derivative of a Vector Field
Let $X,Y$ be vector fields on a smooth manifold $M$ and let $\phi_t$ be the flow of $X$.

I was trying to write the lie derivative of $Y$ w.r.t. $X$ in local co-ordinates, so I have that

$\phi_{-t}^* \left( \frac{\partial}{\partial x_j} \Big|_{\phi_t(p)} \right) = \displaystyle\sum_{i,j} \frac{\partial \phi_i}{\partial x_j} (-t, \phi_t (p)) \frac{\partial}{\partial x_j} \Big|_p$.

I got this because I know that

$\phi_t^* \left( \frac{\partial}{\partial x_j} \Big|_p \right) = \displaystyle\sum_{i,j} \frac{\partial \phi_ i}{\partial x_j} (t,p) \frac{\partial}{\partial x_j} \Big|_{\phi_t(p)},$

So I just swapped $p$ for $\phi_t(p)$ and $t$ for $-t$.

But I have been told this calculation should give

$\phi_{-t}^* \left( \frac{\partial}{\partial x_j} \Big|_{\phi_t(p)} \right) = \displaystyle\sum_{i,j} \frac{\partial \phi_i}{\partial x_j} (-t, p) \frac{\partial}{\partial x_j} \Big|_p$, and it has to for various proofs to work.

Can anyone tell me why I am wrong? I would appreciate it. Thanks.
• November 14th 2011, 03:01 PM
slevvio
Re: Lie Derivative of a Vector Field
The stars should be on the bottom and represent pushforwards