The unitary connections on the complex Grassmann manifold

In the complex Grassmann manifold F(m, n), the space of complex n-planes passes through the origin of Cm+n; the local coordinate of the space can be arranged into an m x n matrix Z. It is proved that K = K(Z, dZ) = (I + ZZ dagger)(-1/2)partial derivative(I + ZZ dagger)(1/2) - partial derivative(I + ZZ dagger)(1/2).(I + ZZ dagger)(-1/2) is a U(m)-connection of F(m, n) and its curvature form Omega(1) = dK + K boolean AND K satisfies the Yang-Mills equation. Moreover, B = B(Z,dZ) = K(Z,dZ)- tr(K(Z,dZ))/mI(m) is an SU(m)-connection and its curvature form Omega(2)= dB + B boolean AND B satisfies the Yang-Mills equation.