Let $E\to X$ and $F\to X$ be complex vector bundles and $\alpha:E\to F$ a bundle isomorphism. Let $\nabla^F$ be a connection on $F\to X$. Then $\alpha^*\nabla^F:=\alpha^{-1}\circ\nabla^F\circ\alpha$ is a connection on $E\to X$. (I know that this is not a standard notation. Do not confuse it with the pullback connection)
I somehow remember that $\text{ch}(\alpha^*\nabla^F)=\text{ch}(\nabla^F)$ (I could be wrong...), where $\text{ch}(\nabla^F)$ is the Chern character form of $\nabla^F$. I couldn't find a reference nor remember (or figure out) why. So if you know a reference or an easy proof, please let me know.
oh my, I simply forgot one of the most important properties of trace, which is $tr(P^{-1}AP)=tr(A)$.