Covariantly constant vector Higgs field ($D_{\mu }\Phi =0$)

If $D_{\mu }\Phi =0$ in the SO(3) vector Higgs model, antisymmetry of the gauge potential implies ${\partial }_{\mu }|\Phi {|}^2=0$. On a local patch with nonzero constant norm, unitary gauge aligns the field with a fixed axis. The connection and gauge field strength then take values in the axis stabilizer; curvature can remain nonzero. Global alignment can require several bundle patches.