Loring W. Tu

Differential Geometry Connections, Curvature, and Characteristic Classes

~Springer

Contents

Preface V

Chapter 1 Curvature and Vector Fields 1

§1 Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1 Inner Produets on a Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Representations of Inner Produets by

Symmetrie Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Riemannian Metries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Existenee of a Riemannian Metrie . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Problems... . ... . ....... . ............ . .. . ............... . ...... 7

§2 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Regular Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Are Length Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Signed Curvature of a Plane Curve . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Orientation and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

§3 Surfaces in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1 Prineipal, Mean, and Gaussian Curvatures. . . . ...... . .... . . . . . 17 3.2 Gauss's Theorema Egregium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 The Gauss-Bonnet Theorem.................. . .. . .... .. .. . 20 Problems....... . . . ........... . ................... . ... . .. . ... . . 21

§4 Directional Derivatives in Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . 22 4.1 Direetional Derivatives in Euelidean Spaee . . . . . . . . . . . . . . . . . . . 22 4.2 Other Properties of the Direetional Derivative . . . . . . . . . . . . . . . . . 24

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4.3 Vector Fields Along a Curve ........................... „ . . 25 4.4 Vector Fields Along a Submanifold ... „ .... „ ... „ ... „ . . . . 26 4.5 Directional Derivatives on a Submanifold of lRn . . . . . . . . . . . • . . . 27 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

§5 The Shape Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.1 Normal Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.2 The Shape Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3 Curvature and the Shape Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.4 The First and Second Fundamental Forms . . . . . . . . . . . . . . . . . . . . 35 5.5 The Catenoid and the Helicoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

§6 Affine Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.1 Affine Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.2 Torsion and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.3 The Riemannian Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.4 Orthogonal Projection on a Surface in JR.3 • • • • . • . • . • . • . . • . . • • • 46 6.5 The Riemannian Connection on a Surface in JR.3 •••.....••..... 47 Problems ...................................................... 48

§7 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7 .1 Definition of a Vector Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7 .2 Tue Vector Space of Sections .... „ .. „ .. „ ....... „ .... „ . 51 7.3 Extending a Local Section to a Global Section . . . . . . . . . . . . . . . . 52 7.4 Local Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 7 .5 Restriction of a Local Operator to an Open Subset . . . . . . . . . . . . . 54 7.6 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7.7 :t-Linearity and Bundle Maps ...... .. ............. „ „ .. „. 56 7 .8 Multilinear Maps over Smooth Functions . . . . . . . . . . . . . . . . . . . . 59

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

§8 Gauss's Theorema Egregium.................................... 61 8.1 The Gauss and Codazzi-Mainardi Equations . . . . . . . . . . . . . . . . . 61 8.2 A Proofof the Theorema Egregium „ ... „ ...... „ •. „ . . . . • . 63 8.3 The Gaussian Curvature in Terms

of an Arbitrary Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

§9 Generalizations to Hypersurfaces in JRn+l • . . . . . • . . . . . • . • . • . . . • . • • 66 9.1 The Shape Operator of a Hypersurface ................. „ . . . • 66 9.2 The Riemannian Connection of a Hypersurface . . . . . . . . . . . . . . . 67 9.3 The Second Fundamental Form.................. ... ........ 68 9.4 The Gauss Curvature and Codazzi-Mainardi Equations . . . . . . . . 68

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Chapter 2 Curvature and Differential Forms 71

§10 Connections on a Vector Bundle.. . . . . ...... .. . . . .... . .. . ... ... .. 71 10.1 Connections on a Vector Bundle .. .. „...................... 72 10.2 Existence of a Connection on a Vector Bundle . . . . . . . . . . . . . . . . 73 10.3 Curvature of a Connection on a Vector Bundle . . . . . . . . . . . . . . . . 74 10.4 Riemannian Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 10.5 Metric Connections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 10.6 Restricting a Connection to an Open Subset . . . . . . . . . . . . . . . . . . 76 10.7 Connections at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

§11 Connection, Curvature, and Torsion Forms..... .................. 79 11.1 Connection and Curvature Forrns .... „ „ . . . . . . . . . . . . . . . . . . . 79 11.2 Connections on a Framed Open Set . „ ........ „ . . . • . . . . . . • . 81 11.3 TheGram-SchmidtProcess ..................... . .... . ..... 81 11.4 Metric Connection Relative to an

Orthonormal Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 11.5 Connections on the Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . . 84 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

§12 The Theorema Egregium Using Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 12.1 The Gauss Curvature Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 12.2 The Theorema Egregium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 12.3 Skew-Symmetries ofthe Curvature Tensor . . . . . . . . . . . . . . . . . . . 91 12.4 Sectional Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 12.5 Poincare Half-Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Chapter 3 Geodesics 95

§13 More on Affine Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 13.1 Covariant Differentiation Along a Curve . . . . . . . . . . . . . . . . . . . . . 95 13.2 Connection-Preserving Diffeomorphisms . . . . . . . . . . . . . . . . . . . . 98 13.3 Christoffel Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Problems ...................................................... 102

§14 Gilodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 14.1 The Definition of a Geodesic .. . „ .•........................ 103 14.2 Reparametrization of a Geodesic ... „ .... ..... ....... ..... . 105 14.3 Existence ofGeodesics ....... . .. .......... ... ............ . 106 14.4 Geodesics in the Poincare Half-Plane ....... ................. 108 14.5 Parallel Translation ..... „ „ ...... „ .............. „ .. „ .. 110

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14.6 Existence of Parallel Translation Along a Curve ............... 111 14.7 Parallel Translation on a Riemannian Manifold ............... 112 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

§15 Exponential Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 15.1 The Exponential Map ofa Connection ... ..... ............... 115 15.2 The Differential ofthe Exponential Map ..................... 117 15.3 Normal Coordinates ...................................... 118 15.4 Left-Invariant Vector Fields on a Lie Group .................. 119 15.5 Exponential Map for a Lie Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 15.6 Naturality of the Exponential Map for a Lie Group . . . . . . . . . . . . 122 15.7 Adjoint Representation ................... ... .............. 123 15.8 Associativity of a Bi-Invariant Metric on a Lie Group .......... 124 Problems ...................................................... 125

15.9 Addendum. The Exponential Map as a Natural Transformation ................................... 126

§16 Distance and Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 16.1 Distance in a Riemannian Manifold . . . . . . . . . . . . . . . . . . . . . . . . . 128 16.2 Geodesic Completeness ................................... 130 16.3 Dual 1-Forms Under a Change ofFrame ..................... 131 16.4 Volume Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 16.5 Tue Volume Form in Local Coordinates. . . . . . . . . . . . . . . . . . . . . . 134 Problems ........ .. ............ ............ ... ................. 135

§17 The Gauss-Bonnet Theorem ........ ............... ... „ ......... 138 17.1 Geodesic Curvature . .......................... ........... 138 17 .2 The Angle Function Along a Curve ........... „ ... ...... „ . 139 17.3 Signed Geodesic Curvature on an Oriented Surface ............ 139 17.4 Gauss-Bonnet Formula for a Polygon .. ........ ..... . ....... 142 17.5 Triangles on a Riemannian 2-Manifold ...................... 144 17.6 Gauss-Bonnet Theorem for a Surface ....................... 145 17.7 Gauss-Bonnet Theorem for a Hypersurface in JR.211+1 •••••••••• 147 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Chapter 4 Tools from Algebra and Topology 151

§18 The Tensor Product and the Dual Module ........................ 151 18.1 Construction ofthe Tensor Product. ......................... 152 18.2 Universal Mapping Property for Bilinear Maps ................ 153 18.3 Characterization ofthe Tensor Product ...................... 154 18.4 A Basis for the Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 18.5 The Dual Module . „ . . . . . . . . . . . • • . . . . . . . . . . . • . • . . . . . • . • • . 157 18.6 Identities for the Tensor Product ............................ 158

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18.7 Functoriality ofthe Tensor Product. . .. ........... . ..... .. ... 160 18.8 Generalization to Multilinear Maps .................... . .... 161 18.9 Associativity of the Tensor Product ......................... 161 18.10 The Tensor Algebra . .. . .... ..... .. ......... . ............. 162 Problems ...................................................... 163

§19 The Exterior Power ..... . ..................................... . 164 19.1 The Exterior Algebra ..................................... 164 19.2 Properties ofthe Wedge Product .. ..................... „ ... 164 19.3 Universal Mapping Property for Alternating

k-Linear Maps ...................................... . .... 166 19.4 A Basis for AkV ......................................... 167 19.5 Nondegenerate Pairings .............. .......... ......... .. 169 19.6 A Nondegenerate Pairing of Ak(Vv) with AkV ............... 170 19.7 A Forrnula for the Wedge Product .......................... 172 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

§20 Operations on Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 20.1 Vector Subbundles .. ................... ......... ......... 174 20.2 Subbundle Criterion ...................................... 175 20.3 Quotient Bundles .... .. .............. ....... .......... ... 176 20.4 The Pullback Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 20.5 Examples ofthe Pullback Bundle ........................... 180 20.6 The Direct Sum ofVector Bundles .......................... 181 20.7 Other Operations on Vector Bundles ........................ 183 Problems .. . ..... .... ... ................... ................ . . .. 185

§21 Vector-Valued Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 21.1 Vector-Valued Forrns as Sections of a Vector Bundle . . . . . . . . . . . 186 21.2 Products ofVector-Valued Forrns ..... .. .................... 188 21.3 Directional Derivative of a Vector-Valued Function . . . . . . . . . . . . 190 21.4 Exterior Derivative of a Vector-Valued Form .................. 190 21.5 Differential Forms with Values in a Lie Algebra ............... 191 21.6 Pullback of Vector-Valued Forrns . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 21.7 Forms with Values in a Vector Bundle .......... . ............ 194 21.8 Tensor Fields on a Manifold ............................... 195 21.9 The Tensor Criterion ........... . ..... . .................. „ 196 21.10 Remark on Signs Concerning Vector-Valued Forrns ............ 197 Problems ............... ....... ............ ... ............. . ... 197

Chapter 5 Vector Bundles and Characteristic Classes 199

§22 Connections and Curvature Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 22.1 Connection and Curvature Matrices Under a Change of Frame. . . 201 22.2 Bianchi Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

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22.3 The First Bianchi Identity in Vector Form . . . . . . . . . . . . . . . . . . . . 204 22.4 Symmetry Properties of the Curvature Tensor . . . . . . . . . . . . . . . . . 205 22.5 Covariant Derivative of Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . 206 22.6 The Second Bianchi ldentity in Vector Form . . . . . . . . . . . . . . . . . . 207 22.7 Ricci Curvature .......................................... 208 22.8 Scalar Curvature ................... . ..... . ............... 209 22.9 Defining a Connection Using Connection Matrices . . . . . . . . . . . . 209 22.10 Induced Connection on a Pullback Bundle ................... 210 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

§23 Cbaracteristic Classes ................ . .......... . . . .. . ......... 212 23.1 Invariant Polynomials on gl(r, IR) ............ . .............. 212 23.2 The Chern-Weil Homomorphism ............... . ........... 213 23.3 Characteristic Forms Are Closed . .. . ............ .. .. . ..... . 215 23.4 Differential Forms Depending on a Real Parameter .. . ... .. .... 216 23.5 Independence of Characteristic Classes of a Connection ........ 218 23.6 Functorial Definition of a Characteristic Class . . . . . . . . . . . . . . . . 220 23.7 Naturality . . ... . ......................................... 221 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

§24 Pontrjagin Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 24.1 Vanishing of Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . . . . 223 24.2 Pontrjagin Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 24.3 The Whitney Product Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

§25 The Euler aass and Chem Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 25.1 Orientation on a Vector Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 25.2 Characteristic Classes of an Oriented Vector Bundle ..... . ..... 229 25.3 The Pfaffian of a Skew-Symmetric Matrix . . . . . . . . . . . . . . . . . . . . 230 25.4 The Euler Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 25.5 Generalized Gauss-Bonnet Theorem ..... . . ... . . ....... .. ... 233 25.6 Hermitian Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 25.7 Connections and Curvature on a Complex

Vector Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 25.8 Chern Classes ................. . ....... . ... . .... . ........ 235 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

§26 Some Applications of Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . 236 26.1 The Generalized Gauss-Bonnet Theorem . . . . . . . . . . . . . . . . . . . . 236 26.2 Characteristic Numbers . ..... . ... .. ............. .. .. . ..... 236 26.3 The Cobordism Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 26.4 The Embedding Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 26.5 The Hirzebruch Signature Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 238 26.6 The Riemann-Roch Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

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Chapter 6 Principal Bundles and Characteristic Classes 241

§27 Principal Bundles ...... „ „ ........... „ •• „ ... „. „ „ „ ....... 241 27.1 Principal Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 27 .2 The Frame Bundle of a Vector B undle . . . . . . . . . . . . . . . . . . . . . . . 246 27.3 Fundamental Vector Fields of a Right Action ................. 247 27.4 Integral Curves of a Fundamental Vector Field . . . . . . . . . . . . . . . . 249 27.5 Vertical Subbundle of the Tangent Bundle T P. . . . . . . . . . . . . . . . . 250 27.6 Horizontal Distributions on a Principal Bundle ................ 251 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

§28 Connections on a Principal Bundle ... „ .. „ ........ „ . „ ....• „ . 254 28.1 Connections on a Principal Bundle .......................... 254 28.2 Vertical and Horizontal Components

of a Tangent Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 28.3 The Horizontal Distribution of an Ehresmann Connection . . . . . . . 257 28.4 Horizontal Lift of a Vector Field to a Principal Bundle . . . . . . . . . 259 28.5 Lie Bracket of a Fundamental Vector Field . . . . . . . . . . . . . . . . . . . 260 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

§29 Horizontal Distributions on a Frame Bundle . . . . . . . . . . . . . . . . . . . . . . 262 29.1 Parallel Translation in a Vector Bundle . .. ................... 262 29.2 Horizontal Vectors on a Frame Bundle ....................... 264 29.3 Horizontal Lift of a Vector Field to a Frame Bundle ............ 266 29.4 Pullback of a Connection on a Frame Bundle Under a Section . . . 268

§30 Curvature on a Principal Bundle . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . 270 30.1 Curvature Form on a Principal Bundle ....................... 270 30.2 Properties of the Curvature Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4

§31 Covariant Derivative on a Principal Bundle ... , . . . . . . . . . . . . . . . . . . . 275 31.1 The Associated Bundle ... „ .... „. „ ... „. „ .... „ .... . ... 275 31.2 The Fiber of the Associated Bundle . „ ....... „ •... „ .. „ „ . 276 31.3 Tensorial Forms on a Principal Bundle ....................... 277 31.4 Covariant Derivative .... „ „ •. „ „ ...... „ . . . . . . . . . . . . . . . . 280 31.5 A Formula for the Covariant Derivative of a

Tensorial Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

§32 Characteristic Classes of Principal Bundles . . . . . . . . . . . . . . . . . . . . . . . 287 32.1 Invariant Polynomials on a Lie Algebra . . . . . . . . . . . . . . . . . . . . . . 287 32.2 The Chern-Weil Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Problems ...................................................... 291

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Appendix 293

§A Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 A.1 Manifolds and Smooth Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 A.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 A.3 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 A.4 Differential Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 A.5 Exterior Differentiation on a Manifold . . . . . . . . . . . . . . . . . . . . . . . 299 A.6 Exterior Differentiation on JR3 . • . • • . • • • . . . • • . • • • • • • . . . • . • . • • 302 A.7 Pullback of Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

§B Invariant Polynomials ......... „ ..... „ ................ „ . . . . . . 306 B.1 Polynomials Versus Polynomial Functions . . . . . . . . . . . . . . . . . . . 306 B.2 Polynomial Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 B.3 Invariant Polynomials on gC(r,F) ........................... 308 B.4 Invariant Complex Polynomials ............................ 310 B.5 L-Polynomials, Todd Polynomials, and the Chern

Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 B.6 Invariant Real Polynomials ................. „ ......... „ „ 315 B.7 Newton's Identities ....................................... 317 Problems ...................................................... 319

Hints and Solutions to Selected End-of-Section Problems 321

List of Notations 329

References 335

Index 337