Curvature difference formula
= Curvature difference formula
{title2=$F_{\nabla+a}=F_\nabla+d_\nabla a+a\wedge a$}
If two objects $\nabla_1$ and $\nabla_2$ are each a <connection on a vector bundle> and satisfy $\nabla_1=\nabla_2+a$ for an $\operatorname{End}(E)$-valued one-form $a$, then
$$
F_{\nabla_1}=F_{\nabla_2}+d_{\nabla_2}a+a\wedge a.
$$
This follows by expanding $(\nabla_2+a)^2$ with the <covariant exterior derivative> on the <endomorphism bundle connection>.