I'm trying to understand the proof of the statement which says that up to multiplication by a nonzero scalar, there is a unique left invariant volume form on a lie group G. What I don't understand in this proof is what is $L_g^*\omega_e$, Where $L_g$ is the left translation by $g \in G$ and $\omega_e$ is $\wedge^n(T_eG)^*$. I know what is the pullback of differential form by a map but in this case $\omega_e$ is not a differential form on G it's an alternating multilinear map. So in general if V is a vector space of dimension n and $\phi : V \rightarrow V $ is a map and $T \in \wedge^n V^*$ what is the definition of $\phi^*(T)$?
Thank you!
Best Answer
The pullback of $T \in \bigwedge^nW^*$ by a linear map $\phi : V \to W$ is defined by
$$(\phi^*T)(v_1, \dots, v_n) = T(\phi(v_1), \dots, \phi(v_n)).$$
Once you have this definition, you can define the pullback of differential forms pointwise. That is, if $f : M \to N$ is a smooth map, we obtain a linear map $(f_*)_p : T_pM \to T_{f(p)}N$, so if $\alpha \in \Omega^n(N)$, we define $(f^*\alpha)_p := (f_*)^*_p\alpha_{f(p)}$, i.e.
$$(f^*\alpha)_p(v_1, \dots, v_n) := ((f_*)^*_p\alpha_{f(p)})(v_1, \dots, v_n) = \alpha_{f(p)}((f_*)_pv_1, \dots, (f_*)_pv_n)$$
for all $v_1, \dots, v_n \in T_pM$.