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I am currently studying the book "Lectures on Kähler geometry" by Andrei Moroianu and am looking for help concerning Exercise 5.8 (3) which is to prove the following Lemma 5.11

Let $f: M \rightarrow N$ be a smooth map. If $E \rightarrow N$ is a vector bundle, then $Gl(f^{*}E) = f^{*}(Gl(E))$. Furthermore, if $P \rightarrow N$ is a $G$-structure and $\rho$ is a representation of $G$ on some vector space $V$, then $f^{*}(P \times_{\rho} V) = (f^{*} P) \times_{\rho} V$.

The following fundamental definitions are given in the book:

Let $f: M \rightarrow N$ be a smooth map between manifolds and let $\pi: P \rightarrow N$ be a $G$-principal bundle over $N$. The $pull-back$ of $P$ by $f$ is defined by $f^{*} P = \{(u, x) \in P \times M \mid \pi(u) = f(x)\}$. It follows from local triviality of $P$ that the map $f^{*} P \rightarrow M, (u, x) \mapsto x$ is a $G$-principal bundle over $M$. The fibre $(f^{*}P)_x$ is canonically identified with the fibre $P_{f(x)}$ by the map $(u, x) \mapsto u$. Moreover, one defines $T_{(u, x)} f^{*} P = \{(V, X) \in T_u P \times T_x M \mid \pi_{*}(V) = f_{*}(X)\}$.

Definition of the frame bundle: Let $M$ be an $n$-dim manifold. A frame at $m \in M$ is a linear isomorphism $u: \mathbb{R}^n \rightarrow T_m M$. The set of all frames at $m$ is denoted by $Gl_m(M)$. $Gl(M)$ denotes the union of all $Gl_m(M)$.

I have already asked this question on Mathematics Stack Exchange. However, I surprisingly received neither an answer nor a comment. For this reason, I dare to ask this question on the site for professional mathematicians. It would be interesting to know from you experts how to assess the level of difficulty of Moroianu´s book and wether questions concerning some of its exercises are acceptable on this site or not. Unfortunately, I could not find any support so far.

Thanks for your help.

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