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

Projective Differential Geometry Old and New

From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups


Published: 1st November 2010
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Ideas of projective geometry keep reappearing in seemingly unrelated fields of mathematics. The authors' main goal is to emphasize connections between classical projective differential geometry and contemporary mathematics and mathematical physics. They also give new results and new 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.

'... 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

Preface: why projective?
The geometry of the projective line
The algebra of the projective line and cohomology of Diff(S1)
Vertices of projective curves
Projective invariants of submanifolds
Projective structures on smooth manifolds
Multi-dimensional Schwarzian derivatives and differential operators
Five proofs of the Sturm theorem
The language of symplectic and contact geometry
The language of connections
The language of homological algebra
Remarkable cocycles on groups of diffeomorphisms
The Godbillon-Vey class
The Adler-Gelfand-Dickey bracket and infinite-dimensional Poisson geometry
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780521831864
ISBN-10: 0521831865
Series: Cambridge Tracts in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 262
Published: 1st November 2010
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.9 x 15.2  x 1.9
Weight (kg): 0.56