= Scalar-curvature criterion for a flat endomorphism connection
The <endomorphism bundle connection> of a rank-$r>0$ bundle is flat if and only if the original curvature satisfies $F_A=\omega\,\operatorname{id}$. By the <curvature of an endomorphism bundle connection>, flatness means each $F_A(X,Y)$ commutes with every fiber endomorphism. The centre of the full matrix algebra consists of scalar matrices, and $\omega=r^{-1}\operatorname{tr}F_A$ is the required smooth two-form. The original connection can still have nonzero scalar curvature.
Back to article page
