I'm working through this question. I can show the forward direction in (a) but can't show the converse. I have $\delta/\delta t \phi^*_t \mu$ evaluated at t=0 is 0, but I can't see how I conclude from here?
In (b), proving the hint is fine. I have also that $d(i_X\mu)= \mu$. Is this true? I'm not really sure what the RHS actually means, and I have no idea how to get the desired result. Can you point me in the right direction?
NB. This is homework
We know that ${d\over dt} \Phi _t^* \mu= \Phi_t^* L_X\mu$. In this case it is $0$, therefore $\Phi _t^* \mu$ is constant equal to $\mu$.
For b, the Einstein convention is used $\partial X^i\over \partial x^i$ means that you have to sum over all indexes. $L_X\mu= d(\sum _i X^i (-1)^i dx^1...dx^{i-1} dx^{i+1} d x^n)= \sum _i {\partial X^i\over \partial x^i}$