I've encountered with following question while reading Morita's book "Geometry of Differential Forms" (pp.263)
Let $(P,\pi,M,G)$ be a principal G-bundle, $\mathfrak{g}$ be the Lie algebra of the Lie group $G$,
He is trying to define fundamental vector field by showing that there is an isomorphism $$V_u:=T_u(\pi^{-1}(p))\cong\mathfrak{g}\tag{1}$$
Here he uses two claims:
Claim.1:for any $p\in M$,there exist a diffeomorphism $i_p\colon G\to \pi^{-1}(p)$ such that $$i_p(hg)=i_p(h)g\tag{2}$$where $h,g\in G$.
In fact, $i_p=\varphi^{-1}(p,\cdot):G\to \pi^{-1}(p)$ where $\varphi \colon \pi^{-1}(U)\to U\times G$ is any local trivialization on $U$ around $p$. It's quite straight forward to see that $i_p$ is a diffeomorphism and satisfies $(2)$
Claim.2:if $\psi :\pi^{-1}(V)\to V\times G$ is another local trivialization, $j_p=\psi^{-1}(p,\cdot):G\to \pi^{-1}(p)$, then $$j_p=i_p\circ L_g$$ for some $g\in G$
Then he concludes:for any $u\in \pi^{-1}(p)$, there exists an isomorphism $(1)$.
I have several questions here:
1.Can we write down the isomorphism $(1)$ explicitly? Claim.2 seems to be used by letting $(i_p)_\ast$ act on a left invariant vector field on $G$, so is the isomorphism $(1)$ given by $$\mathfrak{g}\to V_u,X\mapsto (i_p)_\ast(X)\arrowvert_u?$$
2.Do we really need property $(2)$, which looks like an equivariant condition?
3.Does the isomorphism $(1)$ depend on $u\in \pi^{-1}(p)$ or does it only depend on $p=\pi(u)\in M$?
So much effort, thanks for any hints or answers, as well as any references!
Best Answer
The isomorphism depends on $u$ because it occurs at the point $u \in \pi^{-1}(p)$. You could say that you simultaneously identify all tangent spaces of each fiber by identifying the fiber with $G$ and then right translating the tangent spaces of $G$ to trivialize the tangent bundle of $G$. The isomorphism indeed does not depend on property (2), since property (1) uniquely determines the diffeomorphism to $G$, up to left translation. We can write down the isomorphism explicitly as you did.