Lecture Notes by Prof. Helmut Eschrig
File : pdf, 2,2 MB, 298 pages
TOC
Introduction
1 Topology
1.1 Basic De¯nitions
1.2 Base of Topology, Metric, Norm
1.3 Derivatives
1.4 Compactness
1.5 Connectedness, Homotopy
1.6 Topological Charges in Physics
2 Manifolds
2.1 Charts and Atlases
2.2 Smooth Manifolds
2.3 Tangent Spaces
2.4 Vector Fields
2.5 Mappings of Manifolds, Submanifolds
2.6 Frobenius’ Theorem
2.7 Examples from Physics
3 Tensor Fields
3.1 Tensor Algebras
3.2 Exterior Algebras
3.3 Tensor Fields and Exterior Forms
3.4 Exterior Di®erential Calculus
4 Integration, Homology and Cohomology
4.1 Prelude in Euclidean Space
4.2 Chains of Simplices
4.3 Integration of Di®erential Forms
4.4 De Rham Cohomology
4.5 Homology and Homotopy
4.6 Homology and Cohomology of Complexes
4.7 Euler’s Characteristic
4.8 Critical Points
4.9 Examples from Physics
5 Lie Groups
5.1 Lie Groups and Lie Algebras
5.2 Lie Group Homomorphisms and Representations
5.3 Lie Subgroups
5.4 Simply Connected Covering Group
5.5 The Exponential Mapping
5.6 The General Linear Group Gl(n;K)
5.7 Example from Physics: the Lorentz Group
5.8 The Adjoint Representation
6 Bundles and Connections
6.1 Principal Fiber Bundles
6.2 Frame Bundles
6.3 Connections on Principle Fiber Bundles
6.4 Parallel Transport and Holonomy
6.5 Exterior Covariant Derivative and Curvature Form
6.6 Fiber Bundles
6.7 Linear and A±ne Connections
6.8 Curvature and Torsion Tensors
6.9 Expressions in Local Coordinates on M
7 Parallelism, Holonomy, Homotopy and (Co)homology
7.1 The Exact Homotopy Sequence
7.2 Homotopy of Sections
7.3 Gauge Fields and Connections on R4
7.4 Gauge Fields and Connections on Manifolds
7.5 Characteristic Classes
7.6 Geometric Phases in Quantum Physics
7.7 Gauge Field Theory of Molecular Physics
8 Riemannian Geometry
8.1 Riemannian Metric
8.2 Homogeneous Manifolds
8.3 Riemannian Connection
8.4 Geodesic Normal Coordinates
8.5 Sectional Curvature
8.6 Gravitation
8.7 Complex, Hermitian and KÄahlerian Manifolds
Compendium
C.1 Basic Algebraic Structures (AS)
C.2 Basic Topological (Analytic) Structures
C.3 Smooth Manifolds
C.4 Topological Groups
C.5 Fiber Bundles
C.6 Basic Geometric Structures
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