I'm trying to prove that principal $U(N)$-bundles are in bijective correspondence to complex vector bundles.
I understand the proof that principal $\text{GL}(N, \mathbb{R})$-bundles are equivalent to real vector bundles. Given a principal $\text{GL}(N, \mathbb{R})$-bundle $P$ on a base $X$, use the defining representation of $\text{GL}(N, \mathbb{R})$ on $\mathbb{R}^{N}$ to define an associated real vector bundle $E$ on $X$. Conversely, the frame bundle recovers the original principal bundle.
Obviously, a similar argument holds in the complex case. Now, let $P$ be a principal $U(N)$-bundle on $X$. I can use the defining representation of $U(N)$ on $\mathbb{C}^{N}$ to get an associated complex vector bundle with structure group $U(N)$. Conversely, let $E$ be a complex vector bundle on $X$. I think the key is that every complex vector bundle admits a Hermitian metric which reduces the structure group from $\text{GL}(N, \mathbb{C})$ to $U(N)$. One can then take the frame bundle which will recover the original principal $U(N)$-bundle.
What I'm confused about is that a complex vector bundle may admit many Hermitian metrics, I believe. Do these all give rise to the same principal unitary frame bundle?
A second related question concerns connections and curvature two-forms. Let $P$ be a principal $U(N)$-bundle and $A$ a connection on $P$ with curvature two-form $F_{A}$. The connection is a Lie algebra-valued one-form and $F_{A} \in \Omega(X, adP)$ is a two-form valued in the real vector space $adP$.
Let $E$ be a complex vector bundle equivalent to $P$ under the correspondence described above. Let $\nabla$ be a connection on $E$ with curvature two-form $F_{\nabla}$. We have $F_{\nabla} \in \Omega(X, \text{End}E)$. By the excellent answer to Principal Bundles, Chern Classes, and Abelian Instantons we should have $F_{A} = F_{\nabla}$. In particular, they determine the same Chern classes.
But I'm confused about the relation between the real vector bundle $adP$ and the complex vector bundle $\text{End}E$. Do we have $\text{End}E \cong adP \otimes \mathbb{C}$? Where of course, $E$ and $P$ are equivalent via the above correspondence. To reiterate, the source of my confusion is $F_{\nabla} \in \Omega(X, \text{End}E)$ and $F_{A} \in \Omega(X, adP)$, with $F_{A} = F_{\nabla}$.