nLab complex connection

Let ξ=(p:EB)\xi = (p:E\to B) be a complex-analytic GG-principal bundle where GG is a complex Lie group. The complex-analytic Ehresmann connection is the analytic field of horizontal subspaces, which is GG-equivariant. One can consider the underlying real principal bundle ξ R\xi_{\mathbf{R}}. The operator of complex structure becomes an automorphism I:ξ Rξ RI:\xi_{\mathbf{R}}\to \xi_{\mathbf{R}} of the smooth real G RG_{\mathbf{R}}-bundle ξ R\xi_{\mathbf{R}}. One can consider the differential

(dI) pT pET pE (d I)_p T_p E\to T_p E

which on each vertical subspace (T p VE) R(T_p^V E)_{\mathbf{R}} is an operator of the complex structure on the fiber.

For a field HH of horizontal subspaces on ξ R\xi_{\mathbf{R}} the following are equivalent:

(i) HH is a connection on ξ\xi

(ii) I *H=HI^* H = H

(iii) H p=(dI) pH pH_p = (d I)_p H_p

One can characterize complex connections also by conditions on a covariant derivative on ξ R\xi_{\mathbf{R}}.

Created on January 30, 2012 at 17:29:39. See the history of this page for a list of all contributions to it.