Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups (Cambridge Tracts in Mathematics, Series Number 165)

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. Cover......Page 1 CAMBRIDGE TRACTS IN MATHEMATICS 165......Page 2 About......Page 3 Projective Differential Geometry Old and New: Fromthe Schwarzian Derivative to the Cohomology of Diffeomorphism Groups......Page 4 Copyright - ISBN: 0521831865......Page 5 Contents......Page 6 Preface: why projective?......Page 10 1.1 Projective space and projective duality......Page 14 1.2 Discrete invariants and configurations......Page 18 1.3 Introducing the Schwarzian derivative......Page 21 1.4 A further example of differential invariants: projective curvature......Page 26 1.5 The Schwarzian derivative as a cocycle of Diff(RP^1)......Page 31 1.6 Virasoro algebra: the coadjoint representation......Page 34 2.1 Invariant differential operators on RP^1......Page 39 2.2 Curves in RPn and linear differential operators......Page 42 2.3 Homotopy classes of non-degenerate curves......Page 48 2.4 Two differential invariants of curves: projective curvature and cubic form......Page 53 2.5 Projectively equivariant symbol calculus......Page 55 3 The algebra of the projective line and cohomology of Diff(S^1)......Page 60 3.1 Transvectants......Page 61 3.2 First cohomology of Diff(S^1) with coefficients in differential operators......Page 65 3.3 Application: geometry of differential operators on RP^1......Page 70 3.4 Algebra of tensor densities on S^1......Page 75 3.5 Extensions of Vect(S^1) by the modules F_λ(S^1)......Page 79 4.1 Classic four-vertex and six-vertex theorems......Page 82 4.2 Ghys’ theorem on zeroes of the Schwarzian derivative and geometry of Lorentzian curves......Page 89 4.3 Barner’s theorem on inflections of projective curves......Page 93 4.4 Applications of strictly convex curves......Page 98 4.5 Discretization: geometry of polygons, back to configurations......Page 103 4.6 Inflections of Legendrian curves and singularities of wave fronts......Page 110 5 Projective invariants of submanifolds......Page 116 5.1 Surfaces in RP^3: differential invariants and local geometry......Page 117 5.2 Relative, affine and projective differential geometry of hypersurfaces......Page 129 5.3 Geometry of relative normals and exact transverse line fields......Page 136 5.4 Complete integrability of the geodesic flow on the ellipsoid and of the billiard map inside the ellipsoid......Page 146 5.5 Hilbert’s fourth problem......Page 154 5.6 Global results on surfaces......Page 161 6.1 Definitions, examples and main properties......Page 166 6.2 Projective structures in terms of differential forms......Page 172 6.3 Tensor densities and two invariant differential operators......Page 174 6.4 Projective structures and tensor densities......Page 177 6.5 Moduli space of projective structures in dimension 2, by V. Fock and A. Goncharov......Page 182 7.1 Multi-dimensional Schwarzian with coefficients in (2, 1)-tensors......Page 192 7.2 Projectively equivariant symbol calculus in any dimension......Page 198 7.3 Multi-dimensional Schwarzian as a differential operator......Page 204 7.4 Application: classification of modules D^2_λ(M) for an arbitrary manifold......Page 207 7.5 Poisson algebra of tensor densities on a contact manifold......Page 210 7.6 Lagrange Schwarzian derivative......Page 218 A.1 Five proofs of the Sturm theorem......Page 227 A.2 The language of symplectic and contact geometry......Page 230 A.3 The language of connections......Page 234 A.4 The language of homological algebra......Page 236 A.5 Remarkable cocycles on groups of diffeomorphisms......Page 239 A.6 The Godbillon–Vey class......Page 242 A.7 The Adler–Gelfand–Dickey bracket and infinite-dimensional Poisson geometry......Page 245 References......Page 249 Index......Page 260 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. 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 Ideas of projective geometry keep reappearing in seemingly unrelated fields of mathematics. This book provides a rapid route for graduate students and researchers to the frontiers of contemporary research in this evergreen subject. Exercises play a prominent role: historical and cultural comments relate the basic notions to a broader context The authors explore connections between classical projective differential geometry & contemporary mathematics & mathematical physics, offering new results & new proofs to classic theorems. Related topics include differential operators, the cohomology of the group of diffeomorphisms of the circle, & more