Geometry of differential forms (Book, 2001) [WorldCat.org]
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Geometry of differential forms

Author: Shigeyuki Morita
Publisher: Providence, R.I. : American Mathematical Society, ©2001.
Series: Translations of mathematical monographs, v. 201.; Iwanami series in modern mathematics.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:

Presents a comprehensive introduction to differential forms. This work begins with a presentation of the notion of differentiable manifolds and then develops basic properties of differential forms as  Read more...

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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Shigeyuki Morita
ISBN: 0821810456 9780821810453
OCLC Number: 46619517
Description: xxiv, 321 pages : illustrations ; 22 cm
Contents: 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.
Series Title: Translations of mathematical monographs, v. 201.; Iwanami series in modern mathematics.
Other Titles: Bibun keishiki no kikagaku.
Responsibility: Shigeyuki Morita ; translated by Teruko Nagase, Katsumi Nomizu.

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