Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201) (Translations of Mathematical Monographs)

This work introduces the theory and practice of differential forms on manifolds and overviews the concept of differentiable manifolds, assuming a minimum of knowledge in linear algebra, calculus, and elementary topology. Chapters cover manifolds, differential forms, the de Rham theorem, Laplacian and harmonic forms, and vector and fiber bundles and characteristic classes. The text includes exercises and answers. First published in Japanese by Iwanami Shoten, Publishers, Tokyo, 1997, 1998. Outline And Goal Of The Theory Xix -- Chapter 1 Manifolds 1 -- (a) N-dimensional Numerical Space R[superscript N] 2 -- (b) Topology Of R[superscript N] 3 -- (c) C[infinity] Functions And Diffeomorphisms 4 -- (d) Tangent Vectors And Tangent Spaces Of R[superscript N] 6 -- (e) Necessity Of An Abstract Definition 10 -- 1.2 Definition And Examples Of Manifolds 11 -- (a) Local Coordinates And Topological Manifolds 11 -- (b) Definition Of Differentiable Manifolds 13 -- (c) R[superscript N] And General Surfaces In It 16 -- (d) Submanifolds 19 -- (e) Projective Spaces 21 -- (f) Lie Groups 22 -- 1.3 Tangent Vectors And Tangent Spaces 23 -- (a) C[infinity] Functions And C[infinity] Mappings On Manifolds 23 -- (b) Practical Construction Of C[infinity] Functions On A Manifold 25 -- (c) Partition Of Unity 27 -- (d) Tangent Vectors 29 -- (e) Differential Of Maps 33 -- (f) Immersions And Embeddings 34 -- 1.4 Vector Fields 36 -- (a) Vector Fields 36 -- (b) Bracket Of Vector Fields 38 -- (c) Integral Curves Of Vector Fields And One-parameter Group Of Local Transformations 39 -- (d) Transformations Of Vector Fields By Diffeomorphism 44 -- 1.5 Fundamental Facts Concerning Manifolds 44 -- (a) Manifolds With Boundary 44 -- (b) Orientation Of A Manifold 46 -- (c) Group Actions 49 -- (d) Fundamental Groups And Covering Manifolds 51 -- Chapter 2 Differential Forms 57 -- 2.1 Definition Of Differential Forms 57 -- (a) Differential Forms On R[superscript N] 57 -- (b) Differential Forms On A General Manifold 61 -- (c) Exterior Algebra 61 -- (d) Various Definitions Of Differential Forms 66 -- 2.2 Various Operations On Differential Forms 69 -- (a) Exterior Product 69 -- (b) Exterior Differentiation 70 -- (c) Pullback By A Map 72 -- (d) Interior Product And Lie Derivative 72 -- (e) Cartan Formula And Properties Of Lie Derivatives 73 -- (f) Lie Derivative And One-parameter Group Of Local Transformations 77 -- 2.3 Frobenius Theorem 80 -- (a) Frobenius Theorem--representation By Vector Fields 80 -- (b) Commutative Vector Fields 82 -- (c) Proof Of The Frobenius Theorem 83 -- (d) Frobenius Theorem--representation By Differential Forms 86 -- (a) Differential Forms With Values In A Vector Space 89 -- (b) Maurer-cartan Form Of A Lie Group 90 -- Chapter 3 De Rham Theorem 95 -- 3.1 Homology Of Manifolds 96 -- (a) Homology Of Simplicial Complexes 96 -- (b) Singular Homology 99 -- (c) C[infinity] Triangulation Of C[infinity] Manifolds 100 -- (d) C[infinity] Singular Chain Complexes Of C[infinity] Manifolds 103 -- 3.2 Integral Of Differential Forms And The Stokes Theorem 104 -- (a) Integral Of N-forms On N-dimensional Manifolds 104 -- (b) Stokes Theorem (in The Case Of Manifolds) 107 -- (c) Integral Of Differential Forms On Chains, And The Stokes Theorem 109 -- 3.3 De Rham Theorem 111 -- (a) De Rham Cohomology 111 -- (b) De Rham Theorem 113 -- (c) Poincare Lemma 116 -- 3.4 Proof Of The De Rham Theorem 119 -- (a) Cech Cohomology 119 -- (b) Comparison Of De Rham Cohomology And Cech Cohomology 121 -- (c) Proof Of The De Rham Theorem 126 -- (d) De Rham Theorem And Product Structure 131 -- 3.5 Applications Of The De Rham Theorem 133 -- (a) Hopf Invariant 133 -- (b) Massey Product 136 -- (c) Cohomology Of Compact Lie Groups 137 -- (d) Mapping Degree 138 -- (e) Integral Expression Of The Linking Number By Gauss 140 -- Chapter 4 Laplacian And Harmonic Forms 145 -- 4.1 Differential Forms On Riemannian Manifolds 145 -- (a) Riemannian Metric 145 -- (b) Riemannian Metric And Differentieal Forms 148 -- (c) *-operator Of Hodge 150 -- 4.2 Laplacian And Harmonic Forms 153 -- 4.3 Hodge Theorem 158 -- (a) Hodge Theorem And The Hodge Decomposition Of Differential Forms 158 -- (b) Idea For The Proof Of The Hodoge Decomposition 160 -- 4.4 Applications Of The Hodge Theorem 162 -- (a) Poincare Duality Theorem 162 -- (b) Manifolds And Euler Number 164 -- (c) Intersection Number 165 -- Chapter 5 Vector Bundles And Characteristic Classes 169 -- 5.1 Vector Bundles 169 -- (a) Tangent Bundle Of A Manifold 169 -- (b) Vector Bundles 170 -- (c) Several Constructions Of Vector Bundles 173 -- 5.2 Geodesics And Parallel Translation Of Vectors 180 -- (a) Geodesics 180 -- (b) Covariant Derivative 181 -- (c) Parallel Displacement Of Vectors And Curvature 183 -- 5.3 Connections In Vector Bundles And 185 -- (a) Connections 185 -- (b) Curvature 186 -- (c) Connection Form And Curvature Form 188 -- (d) Transformation Rules Of The Local Expressions For A Connection And Its Curvature 190 -- (e) Differential Forms With Values In A Vector Bundle 191 -- 5.4 Pontrjagin Classes 193 -- (a) Invariant Polynomials 193 -- (b) Definition Of Pontrjagin Classes 197 -- (c) Levi-civita Connection 201 -- 5.5 Chern Classes 204 -- (a) Connection And Curvature In A Complex Vector Bundle 204 -- (b) Definition Of Chern Classes 205 -- (c) Whitney Formula 207 -- (d) Relations Between Pontrjagin And Chern Classes 208 -- 5.6 Euler Classes 211 -- (a) Orientation Of Vector Bundles 211 -- (b) Definition Of The Euler Class 211 -- (c) Properties Of The Euler Class 214 -- 5.7 Applications Of Characteristic Classes 216 -- (a) Gauss-bonnet Theorem 216 -- (b) Characteristic Classes Of The Complex Projective Space 223 -- (c) Characteristic Numbers 225 -- Chapter 6 Fiber Bundles And Characteristic Classes 231 -- 6.1 Fiber Bundle And Principal Bundle 231 -- (a) Fiber Bundle 231 -- (b) Structure Group 233 -- (c) Principal Bundle 236 -- (d) Classification Of Fiber Bundles And Characteristic Classes 238 -- (e) Examples Of Fiber Bundles 239 -- 6.2 S[superscript 1] Bundles And Euler Classes 240 -- (a) S[superscript 1] Bundle 241 -- (b) Euler Class Of An S[superscript 1] Bundle 241 -- (c) Classification Of S[superscript 1] Bundles 246 -- (d) Defining The Euler Class For An S[superscript 1] Bundle By Using Differential Forms 249 -- (e) Primary Obstruction Class And The Euler Class Of The Sphere Bundle 254 -- (f) Vector Fields On A Manifold And Hopf Index Theorem 255 -- 6.3 Connections 257 -- (a) Connections In General Fiber Bundles 257 -- (b) Connections In Principal Bundles 260 -- (c) Differential Form Representation Of A Connection In A Principal Bundle 262 -- 6.4 Curvature 265 -- (a) Curvature Form 265 -- (b) Weil Algebra 268 -- (c) Exterior Differentiation Of The Weil Algebra 270 -- 6.5 Characteristic Classes 275 -- (a) Weil Homomorphism 275 -- (b) Invariant Polynomials For Lie Groups 279 -- (c) Connections For Vector Bundles And Principal Bundles 282 -- (d) Characterisric Classes 284 -- 6.6 A Couple Of Items 285 -- (a) Triviality Of The Cohomology Of The Weil Algebra 285 -- (b) Chern-simons Forms 287 -- (c) Flat Bundles And Holonomy Homomorphisms 287. Shigeyuki Morita ; Translated By Teruko Nagase, Katsumi Nomizu. Includes Bibliographical References (p. 315-316) And Index. Cover ......Page 1 Title page ......Page 2 Date-line ......Page 3 Contents ......Page 4 Preface ......Page 10 Preface to the English Edition ......Page 14 Outline and Goal of the Theory ......Page 16 1 ......Page 22 2 ......Page 23 3 ......Page 24 4 ......Page 25 6 ......Page 27 10 ......Page 31 11 ......Page 32 13 ......Page 34 16 ......Page 37 d) Submanifolds ......Page 40 21 ......Page 42 22 ......Page 43 23 ......Page 44 25 ......Page 46 27 ......Page 48 29 ......Page 50 33 ......Page 54 34 ......Page 55 36 ......Page 57 38 ......Page 59 39 ......Page 60 44 ......Page 65 46 ......Page 67 49 ......Page 70 51 ......Page 72 Summary ......Page 75 55 ......Page 76 57 ......Page 78 c) The exterior algebra ......Page 82 d) Various definitions of differential forms ......Page 87 69 ......Page 90 70 ......Page 91 72 ......Page 93 73 ......Page 94 77 ......Page 98 80 ......Page 101 82 ......Page 103 c) Proof of the Frobenius theorem ......Page 104 d) The Frobenius theorem Representation by differential forms ......Page 107 a) Differential forms with values in a vector space ......Page 110 90 ......Page 111 92 ......Page 113 93 ......Page 114 Chapter 3 The de Rham Theorem ......Page 116 96 ......Page 117 99 ......Page 120 100 ......Page 121 103 ......Page 124 104 ......Page 125 107 ......Page 128 109 ......Page 130 111 ......Page 132 113 ......Page 134 116 ......Page 137 119 ......Page 140 121 ......Page 142 126 ......Page 147 131 ......Page 152 133 ......Page 154 136 ......Page 157 c) Cohomology of compact Lie groups ......Page 158 138 ......Page 159 140 ......Page 161 Exercises ......Page 163 a) Riemannian metric ......Page 166 148 ......Page 169 150 ......Page 171 4.2 Laplacian and harmonic forms ......Page 174 a) The Hodge theorem and the Hodge decomoposition of differential forms ......Page 179 160 ......Page 181 162 ......Page 183 164 ......Page 185 165 ......Page 186 166 ......Page 187 167 ......Page 188 a) The tangent bundle of a manifold ......Page 190 170 ......Page 191 173 ......Page 194 180 ......Page 201 181 ......Page 202 183 ......Page 204 185 ......Page 206 186 ......Page 207 188 ......Page 209 d) Transformation rules of the local expressions for a connection and its curvature ......Page 211 e) Differential forms with values in a vector bundle ......Page 212 193 ......Page 214 b) Definition of Pontrjagin classes ......Page 218 201 ......Page 222 a) Connection and curvature in a complex vector bundle ......Page 225 205 ......Page 226 207 ......Page 228 208 ......Page 229 211 ......Page 232 c) Properties of the Euler class ......Page 235 216 ......Page 237 b) Characteristic classes of the complex projective space ......Page 244 225 ......Page 246 Summary ......Page 249 Exercises ......Page 250 a) Fiber bundle ......Page 252 233 ......Page 254 236 ......Page 257 238 ......Page 259 239 ......Page 260 240 ......Page 261 b) Euler class of an $S^1$ bundle ......Page 262 246 ......Page 267 d) Defining the Euler class for an $S^1$ bundle by using differential forms ......Page 270 254 ......Page 275 255 ......Page 276 a) Connections in general fiber bundles ......Page 278 260 ......Page 281 c) Differential form representation of a connection in a principal bundle ......Page 283 265 ......Page 286 268 ......Page 289 c) Exterior differentiation of the Weil algebra ......Page 291 275 ......Page 296 b) Invariant polynomials for Lie groups ......Page 300 c) Connections for vector bundles and principal bundles ......Page 303 d) Characterisric classes ......Page 305 a) Triviality of the cohomology of the Weil algebra ......Page 306 287 ......Page 308 Summary ......Page 312 Exercises ......Page 313 Perspectives ......Page 316 Chapter 1 ......Page 320 Chapter 2 ......Page 323 Chapter 3 ......Page 326 Chapter 4 ......Page 329 Chapter 5 ......Page 331 Chapter 6 ......Page 332 References ......Page 336 E ......Page 338 5 ......Page 26 15 ......Page 36 24 ......Page 45 101 ......Page 122 9 ......Page 30 43 ......Page 64 234 ......Page 255 20 ......Page 41 120 ......Page 141 58 ......Page 79 97 ......Page 118 17 ......Page 38 50 ......Page 71 154 ......Page 175 63 ......Page 84 14 ......Page 35 171 ......Page 192 276 ......Page 297 196 ......Page 217 45 ......Page 66 98 ......Page 119 232 ......Page 253 235 ......Page 256 74 ......Page 95 198 ......Page 219 226 ......Page 247 frame field ......Page 339 206 ......Page 227 60 ......Page 81 199 ......Page 220 175 ......Page 196 290 ......Page 311 209 ......Page 230 258 ......Page 279 264 ......Page 285 12 ......Page 33 67 ......Page 88 177 ......Page 198 252 ......Page 273 112 ......Page 133 114 ......Page 135 115 ......Page 136 87 ......Page 108 8 ......Page 29 152 ......Page 173 123 ......Page 144 176 ......Page 197 161 ......Page 182 147 ......Page 168 212 ......Page 233 213 ......Page 234 41 ......Page 62 59 ......Page 80 288 ......Page 309 172 ......Page 193 nonzero section ......Page 340 81 ......Page 102 88 ......Page 109 155 ......Page 176 227 ......Page 248 159 ......Page 180 289 ......Page 310 134 ......Page 155 256 ......Page 277 174 ......Page 195 259 ......Page 280 194 ......Page 215 168 ......Page 189 47 ......Page 68 53 ......Page 74 141 ......Page 162 139 ......Page 160 91 ......Page 112 195 ......Page 216 total space ......Page 341 42 ......Page 63 48 ......Page 69 163 ......Page 184 118 ......Page 139 200 ......Page 221 146 ......Page 167 106 ......Page 127 30 ......Page 51 7 ......Page 28 203 ......Page 224 zero section ......Page 342 37 ......Page 58 151 ......Page 172 278 ......Page 299 Since the times of Gauss, Riemann, and Poincaré, one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool in reaching the main goal of global analysis is the theory of differential forms. This book is a comprehensive introduction to differential forms. It begins with a quick presentation of the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results about them, such as the de Rham and Frobenius theorems. The second half of the book is devoted to more advanced material, including Laplacians and harmonic forms on manifolds, the concepts of vector bundles and fiber bundles, and the theory of characteristic classes. Among the less traditional topics treated in the book is a detailed description of the Chern-Weil theory. With minimal prerequisites, the book can serve as a textbook for an advanced undergraduate or a graduate course in differential geometry. Since the times of Gauss, Riemann, and Poincar, one of the principal goals of the study of manifolds has been to relate local analytic properties of a manifold with its global topological properties. Among the high points on this route are the Gauss-Bonnet formula, the de Rham complex, and the Hodge theorem; these results show, in particular, that the central tool in reaching the main goal of global analysis is the theory of differential forms. The book by Morita is a comprehensive introduction to differential forms. It begins with a quick introduction to the notion of differentiable manifolds and then develops basic properties of differential forms as well as fundamental results concerning them, such as the de Rham and Frobenius theorems. The second half of the book is devoted to more advanced material, including Laplacians and harmonic forms on manifolds, the concepts of vector bundles and fiber bundles, and the theory of characteristic classes. Among the less traditional topics treated is a detailed description of the Chern-Weil theory. The book can serve as a textbook for undergraduate students and for graduate students in geometry.