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

ISBN-13: 9780521831864 / Angielski / Twarda / 2004 / 262 str.

Valentin Ovsienko; V. Ovsienko; S. Tabachnikov

Opis

Ideas of projective geometry keep reappearing in seemingly unrelated fields of mathematics. This book provides a rapid route for graduate students and researchers to contemplate the frontiers of contemporary research in this classic subject. The authors include exercises and historical and cultural comments relating the basic ideas to a broader context.

Recenzje

'… 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

Spis treści

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.