A metric compatible connection satisfies:

where c is a christoffel symbol and K is the contortion tensor.

where s is just a place holder.

source psfile jl@crush.caltech.edu index metric_compatible_connection holomorphic_principle_bundle principle_bundle_connection holomorphic_vector_bundle Yang-Mills_gauge_field invariant_polynomial characteristic_class connection_one-form Chern-Weil_theorem spin_connection flat_connection berry_phase derivative holonomy