Invariance and Flows
When we discussed the Lie algebra of a Lie group we discussed “left-invariant” vector fields. More generally than this if 
Now we want a characterization of 
We’ll show this by showing that the vector field 
So, what makes 
More involved is the differential condition. It will help if we rewrite 
\right)\right)\\&=\left[f\circ\Phi\circ\left(1_\mathbb{R}\times f^{-1}\right)\right](t,p)\end{aligned}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Ctilde%7B%5CPhi%7D%28t%2Cp%29%26%3Df%5Cleft%28%5CPhi_t%5Cleft%28f%5E%7B-1%7D%28p%29%5Cright%29%5Cright%29%5C%5C%26%3Df%5Cleft%28%5CPhi%5Cleft%28t%2Cf%5E%7B-1%7D%28p%29%5Cright%29%5Cright%29%5C%5C%26%3Df%5Cleft%28%5CPhi%5Cleft%28%5Cleft%5B1_%5Cmathbb%7BR%7D%5Ctimes+f%5E%7B-1%7D%5Cright%5D%28t%2Cp%29%5Cright%29%5Cright%29%5C%5C%26%3D%5Cleft%5Bf%5Ccirc%5CPhi%5Ccirc%5Cleft%281_%5Cmathbb%7BR%7D%5Ctimes+f%5E%7B-1%7D%5Cright%29%5Cright%5D%28t%2Cp%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}\tilde{\Phi}(t,p)&=f\left(\Phi_t\left(f^{-1}(p)\right)\right)\\&=f\left(\Phi\left(t,f^{-1}(p)\right)\right)\\&=f\left(\Phi\left(\left[1_\mathbb{R}\times f^{-1}\right](t,p)\right)\right)\\&=\left[f\circ\Phi\circ\left(1_\mathbb{R}\times f^{-1}\right)\right](t,p)\end{aligned}")
Now we can start on the differential condition:
![\displaystyle\begin{aligned}\tilde{\Phi}_*\left(\frac{\partial}{\partial t}(t,p)\right)&=f_*\left(\Phi_*\left((1_\mathbb{R}\times f^{-1})_*\left(\iota_{p*}\left(\frac{d}{dt}(t)\right)\right)\right)\right)\\&=f_*\left(\Phi_*\left(\left((1_\mathbb{R}\times f^{-1})\circ\iota_p\right)_*\left(\frac{d}{dt}(t)\right)\right)\right)\\&=f_*\left(\Phi_*\left(\iota_{f^{-1}(p)*}\left(\frac{d}{dt}(t)\right)\right)\right)\\&=f_*\left(X\left(\Phi\left(t,f^{-1}(p)\right)\right)\right)\\&=\left[f_*\circ X\circ\Phi\right]\left(t,f^{-1}(p)\right)\\&=\left[f_*\circ X\circ f^{-1}\circ f\circ\Phi\circ(1_\mathbb{R}\times f^{-1})\right](t,p)\\&=\left[\tilde{X}\circ\tilde{\Phi}\right](t,p)\\&=\tilde{X}\left(\tilde{\Phi}(t,p)\right)\end{aligned}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Ctilde%7B%5CPhi%7D_%2A%5Cleft%28%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+t%7D%28t%2Cp%29%5Cright%29%26%3Df_%2A%5Cleft%28%5CPhi_%2A%5Cleft%28%281_%5Cmathbb%7BR%7D%5Ctimes+f%5E%7B-1%7D%29_%2A%5Cleft%28%5Ciota_%7Bp%2A%7D%5Cleft%28%5Cfrac%7Bd%7D%7Bdt%7D%28t%29%5Cright%29%5Cright%29%5Cright%29%5Cright%29%5C%5C%26%3Df_%2A%5Cleft%28%5CPhi_%2A%5Cleft%28%5Cleft%28%281_%5Cmathbb%7BR%7D%5Ctimes+f%5E%7B-1%7D%29%5Ccirc%5Ciota_p%5Cright%29_%2A%5Cleft%28%5Cfrac%7Bd%7D%7Bdt%7D%28t%29%5Cright%29%5Cright%29%5Cright%29%5C%5C%26%3Df_%2A%5Cleft%28%5CPhi_%2A%5Cleft%28%5Ciota_%7Bf%5E%7B-1%7D%28p%29%2A%7D%5Cleft%28%5Cfrac%7Bd%7D%7Bdt%7D%28t%29%5Cright%29%5Cright%29%5Cright%29%5C%5C%26%3Df_%2A%5Cleft%28X%5Cleft%28%5CPhi%5Cleft%28t%2Cf%5E%7B-1%7D%28p%29%5Cright%29%5Cright%29%5Cright%29%5C%5C%26%3D%5Cleft%5Bf_%2A%5Ccirc+X%5Ccirc%5CPhi%5Cright%5D%5Cleft%28t%2Cf%5E%7B-1%7D%28p%29%5Cright%29%5C%5C%26%3D%5Cleft%5Bf_%2A%5Ccirc+X%5Ccirc+f%5E%7B-1%7D%5Ccirc+f%5Ccirc%5CPhi%5Ccirc%281_%5Cmathbb%7BR%7D%5Ctimes+f%5E%7B-1%7D%29%5Cright%5D%28t%2Cp%29%5C%5C%26%3D%5Cleft%5B%5Ctilde%7BX%7D%5Ccirc%5Ctilde%7B%5CPhi%7D%5Cright%5D%28t%2Cp%29%5C%5C%26%3D%5Ctilde%7BX%7D%5Cleft%28%5Ctilde%7B%5CPhi%7D%28t%2Cp%29%5Cright%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}\tilde{\Phi}_*\left(\frac{\partial}{\partial t}(t,p)\right)&=f_*\left(\Phi_*\left((1_\mathbb{R}\times f^{-1})_*\left(\iota_{p*}\left(\frac{d}{dt}(t)\right)\right)\right)\right)\\&=f_*\left(\Phi_*\left(\left((1_\mathbb{R}\times f^{-1})\circ\iota_p\right)_*\left(\frac{d}{dt}(t)\right)\right)\right)\\&=f_*\left(\Phi_*\left(\iota_{f^{-1}(p)*}\left(\frac{d}{dt}(t)\right)\right)\right)\\&=f_*\left(X\left(\Phi\left(t,f^{-1}(p)\right)\right)\right)\\&=\left[f_*\circ X\circ\Phi\right]\left(t,f^{-1}(p)\right)\\&=\left[f_*\circ X\circ f^{-1}\circ f\circ\Phi\circ(1_\mathbb{R}\times f^{-1})\right](t,p)\\&=\left[\tilde{X}\circ\tilde{\Phi}\right](t,p)\\&=\tilde{X}\left(\tilde{\Phi}(t,p)\right)\end{aligned}")
And thus 
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