Endomorphism vector bundle isomorphic to the adjoint bundle of its frame bundle? Could somebody help me to prove the following isomorphism (in particular what is the isomorphism)?
\begin{equation}
  End(\xi) \cong ad(E_{\xi}) = E_{\xi} \times_{GL(n,\mathbb{R})} \text{Mat}_n(\mathbb{R})
\end{equation}
where $E_{\xi}$ is the frame bundle associated with $\xi$ and $GL(n,\mathbb{R})$ acts on its Lie algebra $\text{Mat}_n(\mathbb{R})$ by conjugation
Thank you very much for your attention!
 A: I change the notations!
If I am not wrong: $E$ is a principal $GL(n,\mathbb{R})$-bundle over a topological space $X$, $Ad(E)$ is the adjoint vector bundle (over $X$) associated to $E$ and $F(E)\equiv F$ is the frame (vector) bundle (over $X$) associated to $E$.
By definition: there exists an open covering $\{U_{\alpha}\}_{\alpha\in A}$ (for $Ad(E)$) of $X$ such that:


*

*$\pi_1^{-1}(U_{\alpha})\stackrel{\varphi_{\alpha}}{\cong}U_{\alpha}\times\mathfrak{gl}(n,\mathbb{R})=U_{\alpha}\times\mathbb{R}^n_n$, $\varphi_{\alpha}$ is a homeomorphism;

*$pr_1\circ\varphi_{\alpha}=\pi_1$ ;

*getting $U_{\alpha\beta}=U_{\alpha}\cap U_{\beta}\neq\emptyset$, the maps:
\begin{equation}
\varphi_{\beta\displaystyle|\pi_1^{-1}(U_{\alpha\beta})}\circ\varphi^{-1}_{\alpha\displaystyle|\pi_1^{-1}(U_{\alpha\beta})}:(P,M)\in U_{\alpha\beta}\times\mathbb{R}^n_n\to(P,Ad(g_{\alpha\beta}(P)^{-1})(M))\in U_{\alpha\beta}\times\mathbb{R}^n_n
\end{equation}
are homeomorphism and the functions $g_{\alpha\beta}:U_{\alpha\beta}\to GL(n,\mathbb{R})$ are the transition functions of $E$.
In the same way: there exists an open covering $\{V_{\alpha}\}_{\alpha\in A}$ (for $End(F)$) of $X$ such that:


*

*$\pi_2^{-1}(V_{\alpha})\stackrel{\psi_{\alpha}}{\cong}V_{\alpha}\times End(\mathbb{R}^n)=V_{\alpha}\times\mathbb{R}^n_n$, $\psi_{\alpha}$ is a homeomorphism;

*$pr_1\circ\psi_{\alpha}=\pi_2$ ;

*getting $V_{\alpha\beta}=V_{\alpha}\cap V_{\beta}\neq\emptyset$, the maps:
\begin{equation}
\psi_{\beta\displaystyle|\pi_2^{-1}(V_{\alpha\beta})}\circ\psi^{-1}_{\alpha\displaystyle|\pi_2^{-1}(V_{\alpha\beta})}:(P,M)\in V_{\alpha\beta}\times\mathbb{R}^n_n\to\left(P,\left(^Tg_{\alpha\beta}^{-1}\otimes g_{\alpha\beta}\right)(P)(M)\right)\in V_{\alpha\beta}\times\mathbb{R}^n_n
\end{equation}
are homeomorphism and the functions $g_{\alpha\beta}:V_{\alpha\beta}\to GL(n,\mathbb{R})$ are the transition functions of $E$ (and of $F$).
Remark. For any pair of vector bundles $V$ and $W$ over $X$: $Hom(V,W)\cong V^{\vee}\otimes W$!, where $V^{\vee}$ is the dual (vector) bundle of $V$.
Whitout loss of generality, we can assume that $Ad(E)$ and $F$ have the same open covering of trivialization $\{U_i\}_{i\in I}$ over $X$!, by a simply computation:
\begin{gather}
\forall i,j\in I,P\in U_{ij}\neq\emptyset,M\in\mathbb{R}^n_n,\\
Ad(g_{ij}^{-1}(P))(M)=g_{ij}^{-1}(P)\times M\times g_{ij}(P)=\left(^Tg_{ij}^{-1}\otimes g_{ij}\right)(P)(M);
\end{gather}
in other words, because $Ad(E)$ and $End(F)$ have the same open covering of trivialization and the same transition functions, they are canonically isomorphic!
A: For a vector bundle $E$, here is a "natural" bijection between its endomorphism bundle $\mathrm{End}(E)$ and its adjoint bundle $F(E) \stackrel{\mathrm{GL}_n}{\times} \mathrm{M}_n$. Each fiber of $\mathrm{End}(E)$ above a point $p$ consists of linear mappings $L:E_p\to E_p$, while the fiber of the adjoint bundle consists of pairs $(\beta, M)$, where $\beta$ is a basis of $E_p$ and $M$ is a matrix.
The bijection sends $L$ to the pair $(\beta, [L]_\beta)$, where $\beta$ is any basis and $[L]_\beta$ is the matrix of $L$ with respect to $\beta$. This is well-defined, because change of basis for an endomorphism is given by the (dual) adjoint representation: that is, if we chose any other basis $A{\cdot} \beta$ for $A\in \mathrm{GL}_n$, we would send $L$ to:
$$
(A{\cdot}\beta, [L]_{A\cdot\beta}) \ =\ (A{\cdot}\beta, A^{-1}[L]_\beta A)
\ \sim\
(\beta, AA^{-1}[L]_\beta AA^{-1}) \ =\ (\beta, [L]_\beta),
$$
namely the same element as before. The inverse of the bijection evidently sends $(\beta, M)$ to the endomorphism $L$ defined by the matrix $M$ with respect to basis $\beta$, and again this is well-defined with respect to the $\mathrm{GL}_n$ quotient because of the change-of-basis formula for matrices.
This is a general principle (and the frame bundle is a general principal): any vector space construction with a bundle corresponds to a vector bundle associated to the frame bundle, by realizing the construction with respect to an arbitrary basis.
