I am reading Taubes's book about differential geometry but I meet some difficulty.
Taubes claims that:Suppose $G$ a matrix Lie group and $P=M \times G$ a principal bundle , then $A=g^{-1}dg$ a connection on $P$. Now consider map $\phi: M\times G \rightarrow M \times G:(x,g)\rightarrow (x,h(x)g)$ .Where $h$ a smooth map from $M$ to $G$.Then the pullback connection $\phi^*A=g^{-1}dg+h^{-1}dh$.
The following is my calculation ,I hope someone can help me to find that what's wrong with it.
Pick $v_1 \in T_xM$ and$v_2 \in T_gG$ ,we have $\phi^*A|_{(x,g)}(v_1,v_2)=A(\phi_*(v_1,v_2))=(h(x)g)^{-1}dh(x)g (\phi_*(v_1,v_2))$.
Since$\phi_*(v_1,v_2)=R_{g*}h_*(v_1)+L_{h*} (v_2)$(here$R_g$ and $L_h$ means that multiple $g$ in the right and multiple $h$ in the left).
Hence $(h(x)g)^{-1}dh(x)g (\phi_*(v_1,v_2))$=$g^{-1}v_2+g^{-1}h^{-1}h_*v_1 g= g^{-1}dg(v_2)+g^{-1}h^{-1}dh(v_1)g$，which is not coincide with the claims above.
Thank you for your help！