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Differential Geometry: Bundles, Connections, Metrics and Curvature [Mīkstie vāki]

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(William Petschek Professor of Mathematics, Harvard University)
  • Formāts: Paperback / softback, 314 pages, height x width x depth: 228x156x27 mm, weight: 483 g, 2 black and white illustrations
  • Sērija : Oxford Graduate Texts in Mathematics 23
  • Izdošanas datums: 13-Oct-2011
  • Izdevniecība: Oxford University Press
  • ISBN-10: 0199605874
  • ISBN-13: 9780199605873
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Differential Geometry: Bundles, Connections, Metrics and CurvaturePalielināt
  • Formāts: Paperback / softback, 314 pages, height x width x depth: 228x156x27 mm, weight: 483 g, 2 black and white illustrations
  • Sērija : Oxford Graduate Texts in Mathematics 23
  • Izdošanas datums: 13-Oct-2011
  • Izdevniecība: Oxford University Press
  • ISBN-10: 0199605874
  • ISBN-13: 9780199605873
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9780199605873.html
Keywords:
Bundles, connections, metrics and curvature are the lingua franca of modern differential geometry and theoretical physics. This book will supply a graduate student in mathematics or theoretical physics with the fundamentals of these objects. Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. Other material covered includes the basic theorems about geodesics and Jacobi fields, the classification theorem for flat connections, the definition of characteristic classes, and also an introduction to complex and Kahler geometry.Differential Geometry uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life. Helpfully, proofs are offered for almost all assertions throughout. All of the introductory material is presented in full and this is the only such source with the classical examples presented in detail.
1 Smooth manifolds
1(13)
1.1 Smooth manifolds
1(2)
1.2 The inverse function theorem and implicit function theorem
3(1)
1.3 Submanifolds of Rm
4(3)
1.4 Submanifolds of manifolds
7(1)
1.5 More constructions of manifolds
8(1)
1.6 More smooth manifolds: The Grassmannians
9(5)
Appendix 1.1 How to prove the inverse function and implicit function theorems
11(2)
Appendix 1.2 Partitions of unity
13(1)
Additional reading
13(1)
2 Matrices and Lie groups
14(11)
2.1 The general linear group
14(1)
2.2 Lie groups
15(1)
2.3 Examples of Lie groups
16(1)
2.4 Some complex Lie groups
17(2)
2.5 The groups Sl(n; C); U(n) and SU(n)
19(2)
2.6 Notation with regards to matrices and differentials
21(4)
Appendix 2.1 The transition functions for the Grassmannians
22(2)
Additional reading
24(1)
3 Introduction to vector bundles
25(14)
3.1 The definition
25(2)
3.2 The standard definition
27(1)
3.3 The first examples of vector bundles
28(1)
3.4 The tangent bundle
29(2)
3.5 Tangent bundle examples
31(2)
3.6 The cotangent bundle
33(1)
3.7 Bundle homomorphisms
34(1)
3.8 Sections of vector bundles
35(1)
3.9 Sections of TM and T*M
36(3)
Additional reading
38(1)
4 Algebra of vector bundles
39(9)
4.1 Subbundles
39(1)
4.2 Quotient bundles
40(1)
4.3 The dual bundle
41(1)
4.4 Bundles of homomorphisms
42(1)
4.5 Tensor product bundles
43(1)
4.6 The direct sum
43(1)
4.7 Tensor powers
44(4)
Additional reading
46(2)
5 Maps and vector bundles
48(11)
5.1 The pull-back construction
48(1)
5.2 Pull-backs and Grassmannians
49(1)
5.3 Pull-back of differential forms and push-forward of vector fields
50(2)
5.4 Invariant forms and vector fields on Lie groups
52(1)
5.5 The exponential map on a matrix group
53(2)
5.6 The exponential map and right/left invariance on Gl(n; C) and its subgroups
55(2)
5.7 Immersion, submersion and transversality
57(2)
Additional reading
58(1)
6 Vector bundles with Cn as fiber
59(13)
6.1 Definitions
59(1)
6.2 Comparing definitions
60(2)
6.3 Examples: The complexification
62(1)
6.4 Complex bundles over surfaces in R3
63(1)
6.5 The tangent bundle to a surface in R3
64(1)
6.6 Bundles over 4-dimensional submanifolds in R5
64(1)
6.7 Complex bundles over 4-dimensional manifolds
65(1)
6.8 Complex Grassmannians
65(3)
6.9 The exterior product construction
68(1)
6.10 Algebraic operations
69(1)
6.11 Pull-back
70(2)
Additional reading
71(1)
7 Metrics on vector bundles
72(6)
7.1 Metrics and transition functions for real vector bundles
73(2)
7.2 Metrics and transition functions for complex vector bundles
75(1)
7.3 Metrics, algebra and maps
75(2)
7.4 Metrics on TM
77(1)
Additional reading
77(1)
8 Geodesics
78(18)
8.1 Riemannian metrics and distance
78(1)
8.2 Length minimizing curves
79(2)
8.3 The existence of geodesics
81(1)
8.4 First examples
82(3)
8.5 Geodesics on SO(n)
85(4)
8.6 Geodesics on U(n) and SU(n)
89(3)
8.7 Geodesics and matrix groups
92(4)
Appendix 8.1 The proof of the vector field theorem
93(1)
Additional reading
94(2)
9 Properties of geodesics
96(8)
9.1 The maximal extension of a geodesic
96(1)
9.2 The exponential map
96(2)
9.3 Gaussian coordinates
98(2)
9.4 The proof of the geodesic theorem
100(4)
Additional reading
103(1)
10 Principal bundles
104(21)
10.1 The definition
104(1)
10.2 A cocycle definition
105(1)
10.3 Principal bundles constructed from vector bundles
106(2)
10.4 Quotients of Lie groups by subgroups
108(2)
10.5 Examples of Lie group quotients
110(3)
10.6 Cocycle construction examples
113(3)
10.7 Pull-backs of principal bundles
116(2)
10.8 Reducible principal bundles
118(1)
10.9 Associated vector bundles
119(6)
Appendix 10.1 Proof of Proposition 10.1
121(3)
Additional reading
124(1)
11 Covariant derivatives and connections
125(14)
11.1 Covariant derivatives
125(1)
11.2 The space of covariant derivatives
126(1)
11.3 Another construction of covariant derivatives
127(1)
11.4 Principal bundles and connections
128(6)
11.5 Connections and covariant derivatives
134(1)
11.6 Horizontal lifts
135(1)
11.7 An application to the classification of principal G-bundles up to isomorphism
136(1)
11.8 Connections, covariant derivatives and pull-back bundles
137(2)
Additional reading
138(1)
12 Covariant derivatives, connections and curvature
139(13)
12.1 Exterior derivative
139(2)
12.2 Closed forms, exact forms, diffeomorphisms and De Rham cohomology
141(2)
12.3 Lie derivative
143(1)
12.4 Curvature and covariant derivatives
144(2)
12.5 An example
146(2)
12.6 Curvature and commutators
148(1)
12.7 Connections and curvature
148(2)
12.8 The horizontal subbundle revisited
150(2)
Additional reading
151(1)
13 Flat connections and holonomy
152(18)
13.1 Flat connections
152(1)
13.2 Flat connections on bundles over the circle
153(2)
13.3 Foliations
155(1)
13.4 Automorphisms of a principal bundle
156(1)
13.5 The fundamental group
157(2)
13.6 The flat connections on bundles over M
159(1)
13.7 The universal covering space
159(1)
13.8 Holonomy and curvature
160(2)
13.9 Proof of the classification theorem for flat connections
162(8)
Appendix 13.1 Smoothing maps
164(2)
Appendix 13.2 The proof of the Frobenius theorem
166(3)
Additional reading
169(1)
14 Curvature polynomials and characteristic classes
170(35)
14.1 The Bianchi Identity
170(1)
14.2 Characteristic forms
171(3)
14.3 Characteristic classes: Part 1
174(1)
14.4 Characteristic classes: Part 2
175(2)
14.5 Characteristic classes for complex vector bundles and the Chern classes
177(2)
14.6 Characteristic classes for real vector bundles and the Pontryagin classes
179(1)
14.7 Examples of bundles with nonzero Chern classes
180(9)
14.8 The degree of the map g → gm from SU(2) to itself
189(1)
14.9 A Chern-Simons form
190(15)
Appendix 14.1 The ad-invariant functions on M(n; C)
190(2)
Appendix 14.2 Integration on manifolds
192(5)
Appendix 14.3 The degree of a map
197(7)
Additional reading
204(1)
15 Covariant derivatives and metrics
205(15)
15.1 Metric compatible covariant derivatives
205(3)
15.2 Torsion free covariant derivatives on T*M
208(2)
15.3 The Levi-Civita connection/covariant derivative
210(1)
15.4 A formula for the Levi-Civita connection
211(1)
15.5 Covariantly constant sections
212(2)
15.6 An example of the Levi-Civita connection
214(2)
15.7 The curvature of the Levi-Civita connection
216(4)
Additional reading
218(2)
16 The Riemann curvature tensor
220(25)
16.1 Spherical metrics, flat metrics and hyperbolic metrics
220(3)
16.2 The Schwarzchild metric
223(1)
16.3 Curvature conditions
224(3)
16.4 Manifolds of dimension 2: The Gauss-Bonnet formula
227(2)
16.5 Metrics on manifolds of dimension 2
229(1)
16.6 Conformal changes
230(2)
16.7 Sectional curvatures and universal covering spaces
232(1)
16.8 The Jacobi field equation
233(3)
16.9 Constant sectional curvature and the Jacobi field equation
236(2)
16.10 Manifolds of dimension 3
238(1)
16.11 The Riemannian curvature of a compact matrix group
239(6)
Additional reading
244(1)
17 Complex manifolds
245(23)
17.1 Some basics concerning holomorphic functions on Cn
246(1)
17.2 The definition of a complex manifold
247(1)
17.3 First examples of complex manifolds
248(3)
17.4 The Newlander-Nirenberg theorem
251(4)
17.5 Metrics and almost complex structures on TM
255(1)
17.6 The almost Kahler 2-form
255(1)
17.7 Symplectic forms
256(1)
17.8 Kahler manifolds
257(1)
17.9 Complex manifolds with closed almost Kahler form
258(1)
17.10 Examples of Kahler manifolds
259(9)
Appendix 17.1 Compatible almost complex structures
261(6)
Additional reading
267(1)
18 Holomorphic submanifolds, holomorphic sections and curvature
268(14)
18.1 Holomorphic submanifolds of a complex manifold
268(1)
18.2 Holomorphic submanifolds of projective spaces
269(2)
18.3 Proof of Proposition 18.2, about holomorphic submanifolds in CPn
271(1)
18.4 The curvature of a Kahler metric
272(3)
18.5 Curvature with no (0, 2) part
275(2)
18.6 Holomorphic sections
277(2)
18.7 Example on CPn
279(3)
Additional reading
281(1)
19 The Hodge star
282(7)
19.1 Definition of the Hodge star
282(1)
19.2 Representatives of De Rham cohomology
283(1)
19.3 A fairy tale
284(1)
19.4 The Hodge theorem
285(1)
19.5 Self-duality
286(3)
Additional reading
287(2)
List of lemmas, propositions, corollaries and theorems 289(2)
List of symbols 291(4)
Index 295
Clifford Henry Taubes is the William Petschek Professor of Mathematics at Harvard University. He is a member of the National Academy of Sciences and also the American Academy of Sciences. He was awarded the American Mathematical Society's Oswald Veblen Prize in 1991 for his work in differential geometry and topology. He was also the recipient of the French Academy of Sciences Elie Cartan Prize in 1993, the Clay Research Award in 2008, the National Academy of Sciences' Mathematics Award in 2008, and the Shaw Prize in Mathematics in 2009.