nLab
complex connection

Let ξ=(p:EB) be a complex-analytic G-principal bundle where G is a complex Lie group. The complex-analytic Ehresmann connection is the analytic field of horizontal subspaces, which is G-equivariant. One can consider the underlying real principal bundle ξ R. The operator of complex structure becomes an automorphism I:ξ Rξ R of the smooth real G R-bundle ξ 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 is an operator of the complex structure on the fiber.

For a field H of horizontal subspaces on ξ R the following are equivalent:

(i) H is a connection on ξ

(ii) I *H=H

(iii) H p=(dI) pH p

One can characterize complex connections also by conditions on a covariant derivative on ξ R.

Created on January 30, 2012 17:31:15 by Zoran Škoda (161.53.130.104)