Conjugate connection (${\overline{D}}_V\overline{s}=\overline{D_{V}s}(V\text{ real})$)

A complex bundle connection induces a connection on its conjugate vector bundle by the displayed rule on real tangent fields. The conjugate scalar action gives the correct Leibniz rule. In conjugate frames the matrix is the complex conjugate of the original connection matrix, with evaluation of one-forms on real tangent vectors understood.