On dual-projectively flat affine connections

Given a pseudo Riemannian metrich and a torsion-free affine connection on a smoothn-manifold M,a dual geodesic curve of is defined as a curve whose tangent 1-form is parallel along the curve. The corresponding dual-projective group is defined as a group of transformations of connections preserving dual-geodesic curves. The class of connections semi-compatible with the metrich and pairs of semi-conjugate connections are defined using the relations between their geodesics and dual-geodesics. The dual-projective curvature tensor for a connection semi-compatible withh is determined as an invariant of the dual projective group. Dual-projectively flat connections semi-compatible withh are characterized as connections with vanishing dual-projective curvature tensor. As an application we recover the fundamental theorem for non-degenerate hypersurface immersions.Research partialy supported by Contract MM 18/1991 with the Ministry of Science and Education of Bulgaria and by Contract with the University of Sofia.