1 - Introduction

Summary

… the field of projective differential geometry is so rich that it seems well worth while to cultivate it with greater energy than has been done heretofore.

E. J. Wilczynski

In this introductory chapter we present a panorama of the subject of this book. The reader who decides to restrict himself to this chapter will get a rather comprehensive impression of the area.

We start with the classical notions of curves in projective space and define projective duality. We then introduce first differential invariants such as projective curvature and projective length of non-degenerate plane projective curves. Linear differential operators in one variable naturally appear here to play a crucial role in what follows.

Already in the one-dimensional case, projective differential geometry offers a wealth of interesting structures and leads us directly to the celebrated Virasoro algebra. The Schwarzian derivative is the main character here. We tried to present classical and contemporary results in a unified synthetic manner and reached the material discovered as late as the last decades of the twentieth century.

Projective space and projective duality

Given a vector space V, the associated projective space, ℙ(V), consists of one-dimensional subspaces of V. If V = ℝⁿ⁺¹ then ℙ(V) is denoted by ℝℙⁿ. The projectivization, ℙ(U), of a subspace U ⊂ V is called a projective subspace of ℙ(V).