PROJECTIVE DIFFERENTIAL GEOMETRY OLD AND NEW
Ouvrage 9780521831864 : PROJECTIVE DIFFERENTIAL GEOMETRY OLD AND NEW
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.
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.
Auteur : OVSIENKO
Editeur : CAMBRIDGE UNIVERSITY PRESS
Nombre de pages : 262
Date de publication : 02 2005
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