Holomorphic projective connections on surfaces and osculating spaces

We study complex analytic projective connections on surfaces in projective n-spaces in terms of the “second” neighborhood of the surface in the ambient space, and in terms of the osculating behavior of the integral curves. We also investigate the action of a remarkable rational transformation on projective connections, and give the geometrical interpretation of joint invariants of a group closely related to the study of equivalence classes of projective connections on surfaces.