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Projective Differential Geometry Old and New

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


Part of Cambridge Tracts in Mathematics

  • Date Published: January 2005
  • availability: Available
  • format: Hardback
  • isbn: 9780521831864

£ 106.00

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About the Authors
  • 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.

    • Presents and summarizes recent research scattered in mathematical journals, and features historical, cultural and bibliographical comments collected at the end of each section
    • Numerous exercises help the reader to master basic techniques and make it possible to avoid lengthy computation in the text of the book
    • A unique feature of this book is that it puts classical projective differential geometry into a broader mathematical context and connects it with contemporary mathematics and mathematical physics
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    Reviews & endorsements

    '… 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

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    Product details

    • Date Published: January 2005
    • format: Hardback
    • isbn: 9780521831864
    • length: 262 pages
    • dimensions: 235 x 160 x 20 mm
    • weight: 0.497kg
    • contains: 53 b/w illus. 35 exercises
    • availability: Available
  • Table of 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

  • Authors

    V. Ovsienko, Université Lyon I

    S. Tabachnikov, Pennsylvania State University

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