Confusion over the definition of Lie Derivative of a Vector Field

[slevvio]

Hello all, I was hoping someone would be able to clarify this issue I am having with the Lie Derivative of a vector field.

We define the lie derivative of a vector field $Y$ with respect to a vector field $X$ to be

$L_X Y :=\operatorname{\frac{d}{dt}} |_{t=0} (\phi_t^*Y)$, where $\phi_t$ is the flow of $X$. Now how do I know this thing is differentiable? And also, I'm not sure why this thing is a vector field on $M$ because what if our flow is only defined on a smaller open set $V \subseteq M$, and then for $p \in M\setminus V$ surely $(L_X Y)_p$ doesn't make sense as a vector in $T_p M$ ?

Anyway thank you, any help would be appreciated.

 

[quasar987]

First concern: You know that phi varies smoothly with t. Go through the definitions to discover that wrt some fixed basis d/dx^i of T_pM, $((\phi_t)_*Y)(p)=\sum_ia_i(t,p)\frac{ \partial}{ \partial x^i}$ with a_i(t,p) smooth in t.

Second concern: the flow of any time-independant vector field is defined at all points p of M for a certain time interval $(-\epsilon_p,\epsilon_p)$

 

[slevvio]

Thanks for your response. I understand the 2nd concern but I still have trouble with the first. I cannot see why that statement is true. I know that $\phi_t ^* Y$ is a vector field, so $\phi_t^* Y |_p \in T_p M$ , i.e.

$\phi_t^* Y |_p = \displaystyle\sum_i a_i^t (p) \frac{\partial}{\partial x_i }\Big|_p$ where for fixed $t$, $p \mapsto a_i^t (p)$ is smooth. But why for fixed $p$, is $t \mapsto a_i^t(p)$ smooth?

I have that $\phi_t^* Y |_p (g) = Y(g \circ \phi_t ^{-1})|_{\phi_t(p)}$ incidentally. Thanks for any help

 

[quasar987]

You're not "going through the definitions" deep enough. What I was suggesting is for you to find an explicit expression for the a_i(t,p).

 

[slevvio]

Hello I have tried to do this using the Jacobian but I am not getting anywhere at all

 

[slevvio]

Hello,

I have managed to calculate that

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

although in Loring Tu's book the same calculation gives

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

Which is correct, and should it matter in the argument?

Thanks for help so far.