Riemannian Holonomy Groups and Calibrated Geometry

Name: Riemannian Holonomy Groups and Calibrated Geometry
Brand: Oxford University Press
Price: 99.49 EUR
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JOYCE DOMINIC D(Autor*in)

Oxford University Press

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Erschienen am 22. Februar 2007

314 Seiten

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978-0-19-152697-8 (ISBN)

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Inhalt

Intro
Contents
Preface
1 Background material
1.1 Exterior forms on manifolds
1.2 Introduction to analysis
1.3 Introduction to elliptic operators
1.4 Regularity of solutions of elliptic equations
1.5 Existence of solutions of linear elliptic equations
2 Introduction to connections, curvature and holonomy groups
2.1 Bundles, connections and curvature
2.2 Vector bundles, connections and holonomy groups
2.3 Holonomy groups and principal bundles
2.4 Holonomy groups and curvature
2.5 Connections on the tangent bundle, and torsion
2.6 G-structures and intrinsic torsion
3 Riemannian holonomy groups
3.1 Introduction to Riemannian holonomy groups
3.2 Reducible Riemannian manifolds
3.3 Riemannian symmetric spaces
3.4 The classification of Riemannian holonomy groups
3.5 Holonomy groups, exterior forms and cohomology
3.6 Spinors and holonomy groups
4 Calibrated geometry
4.1 Minimal submanifolds and calibrated submanifolds
4.2 Calibrated geometry and Riemannian holonomy groups
4.3 Classification of calibrations on R[sup(n)]
4.4 Geometric measure theory and tangent cones
5 Kähler manifolds
5.1 Introduction to complex manifolds
5.2 Tensors on complex manifolds
5.3 Holomorphic vector bundles
5.4 Introduction to Kähler manifolds
5.5 Kähler potentials
5.6 Curvature of Kähler manifolds
5.7 Exterior forms on Kähler manifolds
5.8 Complex algebraic varieties
5.9 Singular varieties, resolutions, and deformations
5.10 Line bundles and divisors
6 The Calabi Conjecture
6.1 Reformulating the Calabi Conjecture
6.2 Overview of the proof of the Calabi Conjecture
6.3 Calculations at a point
6.4 The proof of Theorem C1
6.5 The proof of Theorem C2
6.6 The proof of Theorem C3
6.7 The proof of Theorem C4
6.8 A discussion of the proof
7 Calabi-Yau manifolds
7.1 Ricci-flat Kähler manifolds and Calabi-Yau manifolds
7.2 Crepant resolutions, small resolutions, and flops
7.3 Crepant resolutions of quotient singularities
7.4 Complex orbifolds
7.5 Crepant resolutions of orbifolds
7.6 Complete intersections
7.7 Deformations of Calabi-Yau manifolds
8 Special Lagrangian geometry
8.1 Special Lagrangian submanifolds in C[sup(m)]
8.2 Constructing examples of SL m-folds in C[sup(m)]
8.3 SL cones and Asymptotically Conical SL m-folds
8.4 SL m-folds in (almost) Calabi-Yau m-folds
8.5 SL m-folds with isolated conical singularities
9 Mirror symmetry and the SYZ Conjecture
9.1 String theory and mirror symmetry for dummies
9.2 Early mathematical formulations of mirror symmetry
9.3 Kontsevich's homological mirror symmetry proposal
9.4 The SYZ Conjecture
10 Hyperkähler and quaternionic Kähler manifolds
10.1 An introduction to hyperkähler geometry
10.2 Hyperkähler ALE spaces
10.3 K3 surfaces
10.4 Higher-dimensional compact hyperkähler manifolds
10.5 Quaternionic Kähler manifolds
10.6 Other topics in quaternionic geometry
11 The exceptional holonomy groups
11.1 The holonomy group G[sub(2)]
11.2 Topological properties of compact G[sub(2)]-manifolds
11.3 Constructing compact G[sub(2)]-manifolds
11.4 The holonomy group Spin(7)
11.5 Topological properties of compact Spin(7)-manifolds
11.6 Constructing compact Spin(7)-manifolds
11.7 Further reading on the exceptional holonomy groups
12 Associative, coassociative and Cayley submanifolds
12.1 Associative 3-folds and coassociative 4-folds in R[sup(7)]
12.2 Constructing associative and coassociative k-folds in R[sup(7)]
12.3 Associative 3- and coassociative 4-folds in G[sub(2)]-manifolds
12.4 Cayley 4-folds in R[sup(8)]
12.5 Cayley 4-folds in Spin(7)-manifolds
References
Index
A
B
C
D
E
F
G
H
I
K
L
M
N
O
P
Q
R
S
T
V
W
Y
Z

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