The unitary connections on the complex Grassmann manifold

In the complex Grassmann manifold ?(m,n), the space of complexn-planes passes through the origin of Cm+n; the local coordinate of the space can be arranged into anm ×n matrixZ. It is proved that $$K = K(Z,dZ) = (I + ZZ^\dag )^{ - \frac{1}{2}\overline \partial } (I + ZZ^\dag )^{\frac{1}{2}} - \partial (I + ZZ^\dag )^{\frac{1}{2}} + (I + ZZ^\dag )^{ - \frac{1}{2}} $$ is a U(m)-connection of ?(m,n) and its curvature form $$\Omega _1 = dK + K\Lambda K$$ satisfies the Yang-Mills equation. Moreover, $$B = B(Z,{\bf{ }}dZ) = K(Z,{\bf{ }}dZ) - \frac{{tr(K(Z,{\bf{ }}dZ))}}{m}I^{(m)} $$ is an (Sum)-connection and its curvature form $$\Omega _2 = dB + B{\bf{ }}\Lambda {\bf{ }}B$$ satisfies the Yang-Mills equation.

In the complex Grassmann manifold F(m,n), the space of complex \%n\%\|planes passes through the origin of C \%m+n\% ; the local coordinate of the space can be arranged into an \%m×n\% matrix \%Z.\% It is proved thatK=K(Z,\%d\%Z)=(I+ZZ ) -12 (I+ZZ ) 12 -(I+ZZ ) 12 ·(I+ZZ ) -12 is a \%U(m)\%\|connection of F\%(m,n)\% and its curvature formΩ\-1= d K+K∧Ksatisfies the Yang\|Mills equation. Moreover,B=B(Z, d Z)=K(Z, d Z)- tr (K(Z, d Z))mI (m) is an SU(m) \|connection and its curvature formΩ\-2= d B+B∧Bsatisfies the Yang\|Mills equation.