Abstract
Given a complex Riemannian metrich and a torsion-free complex affine connection ▽ on a complex manifold, a dual holomorphically-planar curve of ▽ is defined as a curve whose tangent complex plane, generated by its tangent 1-form, is parallel along the curve. The corresponding dual holomorphically projective group is defined as a group of transformations of connections preserving dual holomorphically-planar curves. The class of connections complex semi-compatible with the metrich and pairs of complex semi-conjugate connections are defined using the relations between their holomorphically-planar curves and their dual holomorphically-planar curves. The dual holomorphically-projective curvature tensor for a connection complex semi-compatible withh is determined as an invariant of the dual holomorphically-projective group. Dual holomorphically-projectively flat connections complex semi-compatible withh are characterized as connections with vanishing dual holomorphically-projective curvature tensor.
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