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I am stuck in one exercise I found in the book : Topology, geometry and gauge fields.

Here it is : Exercise

Where :

  • $\sigma_g$ is the right action
  • $A^\#$ is the fundamental vector field associated at A.

I manage to show the first term on the equation namely : $(\sigma_{go})_{*s(x_o)}(s_{*x_o}(v_1))$, but I'm stuck with the second term.

My attempt of a solution for the second term: I consider $T_{(x_o,g_o)}(V\times G)$ as $T_{x_o}(V)\times T_{g_o}(G)$ with $(\psi^{-1})_{*(x_o,g_o)}(v_1,v_2) = (\psi_1)_{*x_o}(v_1) + (\psi_2)_{*g_o}(v_2)$. And we have :

\begin{align} &\psi_1:V\rightarrow P^{-1}(V) \\ &\psi_1(x) = \psi^{-1}(x,g_o) = s(x).g_o \\ &\psi_2:G\rightarrow P^{-1}(V) \\ &\psi_2(g) = \psi^{-1}(x_o,g) = s(x_o).g \\ \end{align}

Evertime I try to use this to calculate the push forward of $\psi_2$ I end up nowhere.

I know that : $A^\#(s(x_o).g_o) = (f(s(x_o).g_o .exp(tA)))'(0)$ but I don't know how to inject this in the reasoning.

Could I get some pointer or some results I should be using to try to solve this.

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