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

Since the times of Gauss, Riemann, and Poincare, 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. 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. 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.