Differential Geometry

Name: Differential Geometry | Bundles, Connections, Metrics and Curvature
Brand: OUP eBook
Price: 63.69 EUR
Availability: OnlineOnly

Bundles, Connections, Metrics and Curvature

Clifford Henry Taubes(Autor*in)

OUP eBook (Verlag)

1. Auflage

Erschienen am 14. Oktober 2011

313 Seiten

E-Book

PDF mit Adobe-DRM

Systemvoraussetzungen

978-0-19-162178-9 (ISBN)

63,69 €inkl. 7% MwSt.

Systemvoraussetzungen

für PDF mit Adobe-DRM

E-Book Einzellizenz

Als Download verfügbar

Beschreibung

Weitere Details

Weitere Ausgaben

Person

Inhalt

Cover
Contents
1 Smooth manifolds
1.1 Smooth manifolds
1.2 The inverse function theorem and implicit function theorem
1.3 Submanifolds of R[sup(m)]
1.4 Submanifolds of manifolds
1.5 More constructions of manifolds
1.6 More smooth manifolds: The Grassmannians
Appendix 1.1 How to prove the inverse function and implicit function theorems
Appendix 1.2 Partitions of unity
Additional reading
2 Matrices and Lie groups
2.1 The general linear group
2.2 Lie groups
2.3 Examples of Lie groups
2.4 Some complex Lie groups
2.5 The groups Sl(n
C), U(n) and SU(n)
2.6 Notation with regards to matrices and differentials
Appendix 2.1 The transition functions for the Grassmannians
Additional reading
3 Introduction to vector bundles
3.1 The definition
3.2 The standard definition
3.3 The first examples of vector bundles
3.4 The tangent bundle
3.5 Tangent bundle examples
3.6 The cotangent bundle
3.7 Bundle homomorphisms
3.8 Sections of vector bundles
3.9 Sections of TM and T*M
Additional reading
4 Algebra of vector bundles
4.1 Subbundles
4.2 Quotient bundles
4.3 The dual bundle
4.4 Bundles of homomorphisms
4.5 Tensor product bundles
4.6 The direct sum
4.7 Tensor powers
Additional reading
5 Maps and vector bundles
5.1 The pull-back construction
5.2 Pull-backs and Grassmannians
5.3 Pull-back of differential forms and push-forward of vector fields
5.4 Invariant forms and vector fields on Lie groups
5.5 The exponential map on a matrix group
5.6 The exponential map and right/left invariance on Gl(n
C) and its subgroups
5.7 Immersion, submersion and transversality
Additional reading
6 Vector bundles with C[sup(n)] as fiber
6.1 Definitions
6.2 Comparing definitions
6.3 Examples: The complexification
6.4 Complex bundles over surfaces in R[sup(3)]
6.5 The tangent bundle to a surface in R[sup(3)]
6.6 Bundles over 4-dimensional submanifolds in R[sup(5)]
6.7 Complex bundles over 4-dimensional manifolds
6.8 Complex Grassmannians
6.9 The exterior product construction
6.10 Algebraic operations
6.11 Pull-back
Additional reading
7 Metrics on vector bundles
7.1 Metrics and transition functions for real vector bundles
7.2 Metrics and transition functions for complex vector bundles
7.3 Metrics, algebra and maps
7.4 Metrics on TM
Additional reading
8 Geodesics
8.1 Riemannian metrics and distance
8.2 Length minimizing curves
8.3 The existence of geodesics
8.4 First examples
8.5 Geodesics on SO(n)
8.6 Geodesics on U(n) and SU(n)
8.7 Geodesics and matrix groups
Appendix 8.1 The proof of the vector field theorem
Additional reading
9 Properties of geodesics
9.1 The maximal extension of a geodesic
9.2 The exponential map
9.3 Gaussian coordinates
9.4 The proof of the geodesic theorem
Additional reading
10 Principal bundles
10.1 The definition
10.2 A cocycle definition
10.3 Principal bundles constructed from vector bundles
10.4 Quotients of Lie groups by subgroups
10.5 Examples of Lie group quotients
10.6 Cocycle construction examples
10.7 Pull-backs of principal bundles
10.8 Reducible principal bundles
10.9 Associated vector bundles
Appendix 10.1 Proof of Proposition 10.1
Additional reading
11 Covariant derivatives and connections
11.1 Covariant derivatives
11.2 The space of covariant derivatives
11.3 Another construction of covariant derivatives
11.4 Principal bundles and connections
11.5 Connections and covariant derivatives
11.6 Horizontal lifts
11.7 An application to the classification of principal G-bundles up to isomorphism
11.8 Connections, covariant derivatives and pull-back bundles
Additional reading
12 Covariant derivatives, connections and curvature
12.1 Exterior derivative
12.2 Closed forms, exact forms, diffeomorphisms and De Rham cohomology
12.3 Lie derivative
12.4 Curvature and covariant derivatives
12.5 An example
12.6 Curvature and commutators
12.7 Connections and curvature
12.8 The horizontal subbundle revisited
Additional reading
13 Flat connections and holonomy
13.1 Flat connections
13.2 Flat connections on bundles over the circle
13.3 Foliations
13.4 Automorphisms of a principal bundle
13.5 The fundamental group
13.6 The flat connections on bundles over M
13.7 The universal covering space
13.8 Holonomy and curvature
13.9 Proof of the classification theorem for flat connections
Appendix 13.1 Smoothing maps
Appendix 13.2 The proof of the Frobenius theorem
Additional reading
14 Curvature polynomials and characteristic classes
14.1 The Bianchi Identity
14.2 Characteristic forms
14.3 Characteristic classes: Part 1
14.4 Characteristic classes: Part 2
14.5 Characteristic classes for complex vector bundles and the Chern classes
14.6 Characteristic classes for real vector bundles and the Pontryagin classes
14.7 Examples of bundles with nonzero Chern classes
14.8 The degree of the map g g[sup(m)] from SU(2) to itself
14.9 A Chern-Simons form
Appendix 14.1 The ad-invariant functions on M(n
C)
Appendix 14.2 Integration on manifolds
Appendix 14.3 The degree of a map
Additional reading
15 Covariant derivatives and metrics
15.1 Metric compatible covariant derivatives
15.2 Torsion free covariant derivatives on T*M
15.3 The Levi-Civita connection/covariant derivative
15.4 A formula for the Levi-Civita connection
15.5 Covariantly constant sections
15.6 An example of the Levi-Civita connection
15.7 The curvature of the Levi-Civita connection
Additional reading
16 The Riemann curvature tensor
16.1 Spherical metrics, flat metrics and hyperbolic metrics
16.2 The Schwarzchild metric
16.3 Curvature conditions
16.4 Manifolds of dimension 2: The Gauss-Bonnet formula
16.5 Metrics on manifolds of dimension 2
16.6 Conformal changes
16.7 Sectional curvatures and universal covering spaces
16.8 The Jacobi field equation
16.9 Constant sectional curvature and the Jacobi field equation
16.10 Manifolds of dimension 3
16.11 The Riemannian curvature of a compact matrix group
Additional reading
17 Complex manifolds
17.1 Some basics concerning holomorphic functions on C[sup(n)]
17.2 The definition of a complex manifold
17.3 First examples of complex manifolds
17.4 The Newlander-Nirenberg theorem
17.5 Metrics and almost complex structures on TM
17.6 The almost Kähler 2-form
17.7 Symplectic forms
17.8 Kähler manifolds
17.9 Complex manifolds with closed almost Kähler form
17.10 Examples of Kähler manifolds
Appendix 17.1 Compatible almost complex structures
Additional reading
18 Holomorphic submanifolds, holomorphic sections and curvature
18.1 Holomorphic submanifolds of a complex manifold
18.2 Holomorphic submanifolds of projective spaces
18.3 Proof of Proposition 18.2, about holomorphic submanifolds in CP[sup(n)]
18.4 The curvature of a Kähler metric
18.5 Curvature with no (0, 2) part
18.6 Holomorphic sections
18.7 Example on CP[sup(n)]
Additional reading
19 The Hodge star
19.1 Definition of the Hodge star
19.2 Representatives of De Rham cohomology
19.3 A fairy tale
19.4 The Hodge theorem
19.5 Self-duality
Additional reading
List of lemmas, propositions, corollaries and theorems
List of symbols
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Z

Systemvoraussetzungen

Als PDF speichern Als Link merken