Preface: why projective?

Summary

Metrical geometry is a part of descriptive geometry, and descriptive geometry is all geometry.

Arthur Cayley

On October 5, 2001, the authors of this book typed in the word “Schwarzian” in the MathSciNet database and the system returned 666 hits. Every working mathematician has encountered the Schwarzian derivative at some point of his education and, most likely, tried to forget this rather scary expression right away. One of the goals of this book is to convince the reader that the Schwarzian derivative is neither complicated nor exotic; in fact, it is a beautiful and natural geometrical object.

The Schwarzian derivative was discovered by Lagrange: “According to a communication for which I am indebted to Herr Schwarz, this expression occurs in Lagrange's researches on conformable representation ‘Sur la construction des cartes géographiques’” [117]; the Schwarzian also appeared in a paper by Kummer in 1836, and it was named after Schwarz by Cayley. The main two sources of current publications involving this notion are classical complex analysis and one-dimensional dynamics. In modern mathematical physics, the Schwarzian derivative is mostly associated with conformal field theory. It also remains a source of inspiration for geometers.

The Schwarzian derivative is the simplest projective differential invariant, namely, an invariant of a real projective line diffeomorphism under the natural SL(2,ℝ)-action on ℝℙ.