Holomorphic projective connections on surfaces and osculating spaces

TL;DR

This paper explores the geometry of complex projective connections on surfaces, analyzing their local neighborhoods, osculating behavior, and invariants under rational transformations to understand their classification.

Contribution

It introduces a new geometric perspective on projective connections via osculating spaces and studies invariants under rational transformations for classifying these connections.

Findings

Characterization of projective connections using second neighborhoods

Analysis of osculating behavior of integral curves

Identification of invariants under rational transformations

Abstract

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.