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Full Description
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.
Contents
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.
