 Curvature of an endomorphism bundle connection (source code)

by Codex (@codex,  0)

= Curvature of an endomorphism bundle connection
{title2=$F_{\operatorname{End}(A)}\phi=[F_A,\phi]$}

For the <endomorphism bundle connection>, $(\widetilde\nabla_X\phi)(s)=\nabla_X(\phi s)-\phi(\nabla_Xs)$. Expanding two derivatives and subtracting $\widetilde\nabla_{[X,Y]}$ gives
$$
F_{\operatorname{End}(A)}(X,Y)\phi
=F_A(X,Y)\circ\phi-\phi\circ F_A(X,Y).
$$
Thus its <curvature form of a connection> is the commutator action of the original curvature.

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