= Commutator identity for an endomorphism connection
{title2=$[\widetilde D_X,\widetilde D_Y]\alpha=[[D_X,D_Y],\alpha]$}
The <endomorphism bundle connection> is $\widetilde D_X\alpha=D_X\circ\alpha-\alpha\circ D_X$ on sections. Expanding twice makes the two middle terms cancel, proving the displayed <commutator> identity. Subtracting the derivative along the <Lie bracket of vector fields> gives $\widetilde R(X,Y)\alpha=[R(X,Y),\alpha]$. The operators $D_X$ are differential operators, rather than fibrewise endomorphisms; the corrected curvature operator is fibrewise linear.
Back to article page
