Projective Differential Geometry Old and New
From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups

A rapid route for graduate students and researchers to the frontiers of research in this evergreen subject, first published in 2005.

V. Ovsienko (Author), S. Tabachnikov (Author)

9780521831864, Cambridge University Press

Hardback, published 13 December 2004

262 pages, 53 b/w illus. 35 exercises
23.5 x 16 x 2 cm, 0.497 kg

'… this is an introduction to global projective differential geometry offering felicitous choice of topics, leading from classical projective differential geometry to current fields of research in mathematics and mathematical physics. The reader is guided from simple facts concerning curves and derivatives to more involved problems and methods through a world of inspiring ideas, delivering insights in deep relations. Historical comments as well as stimulating exercises occur frequently throughout the text, making it suitable for teachings.' Zentralblatt MATH

Ideas of projective geometry keep reappearing in seemingly unrelated fields of mathematics. The authors' main goal in this 2005 book is to emphasize connections between classical projective differential geometry and contemporary mathematics and mathematical physics. They also give results and proofs of classic theorems. Exercises play a prominent role: historical and cultural comments set the basic notions in a broader context. The book opens by discussing the Schwarzian derivative and its connection to the Virasoro algebra. One-dimensional projective differential geometry features strongly. Related topics include differential operators, the cohomology of the group of diffeomorphisms of the circle, and the classical four-vertex theorem. The classical theory of projective hypersurfaces is surveyed and related to some very recent results and conjectures. A final chapter considers various versions of multi-dimensional Schwarzian derivative. In sum, here is a rapid route for graduate students and researchers to the frontiers of current research in this evergreen subject.

Preface: why projective?
1. Introduction
2. The geometry of the projective line
3. The algebra of the projective line and cohomology of Diff(S1)
4. Vertices of projective curves
5. Projective invariants of submanifolds
6. Projective structures on smooth manifolds
7. Multi-dimensional Schwarzian derivatives and differential operators
Appendix 1. Five proofs of the Sturm theorem
Appendix 2. The language of symplectic and contact geometry
Appendix 3. The language of connections
Appendix 4. The language of homological algebra
Appendix 5. Remarkable cocycles on groups of diffeomorphisms
Appendix 6. The Godbillon–Vey class
Appendix 7. The Adler–Gelfand–Dickey bracket and infinite-dimensional Poisson geometry
Bibliography
Index.

Subject Areas: Topology [PBP], Geometry [PBM], Algebra [PBF]