On dual holomorphically projectively flat affine connections

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.Research partialy supported by Contract MM 423/1994 with the Ministry of Science and Education of Bulgaria and by Contract 219/1994 with the University of Sofia St. Kl. Ohridski.I would like to thank the referee for indicating omissions in the text and for the helpful advices during the preparation of the final form of the paper.