- #1
- 22,183
- 3,322
- Author: C.J. Isham
- Title: Modern Differential Geometry for Physicists
- Amazon Link: https://www.amazon.com/dp/9810235623/?tag=pfamazon01-20
Table of Contents:
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[*] An Introduction to Topology
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[*] Preliminary Remarks
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[*] Remarks on differential geometry
[*] Remarks on topology
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[*] Metric Spaces
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[*] The simple idea of convergence
[*] The idea of a metric space
[*] Examples of metric spaces
[*] Operations on metrics
[*] Some topological concepts in metric spaces
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[*] Partially Ordered Sets and Lattices
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[*] Partially ordered sets
[*] Lattices
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[*] General Topology
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[*] An example of non-metric convergence
[*] The idea of a neighbourhood space
[*] Topological spaces
[*] Some examples of topologies on a finite set
[*] A topology as a lattice
[*] The lattice of topologies t(X) on a set X
[*] Some properties of convergence in a general topological space
[*] The idea of a compact space
[*] Maps between topological spaces
[*] The idea of a homeomorphism
[*] Separation axioms
[*] Frames and locales
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[*] Differentiable Manifolds
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[*] Preliminary Remarks
[*] The Main Definitions
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[*] Coordinate charts
[*] Some examples of differentiable manifolds
[*] Differentiable maps
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[*] Tangent Spaces
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[*] The intuitive idea
[*] A tangent vector as an equivalence class of curves
[*] The vector space structure on TpM.
[*] The push-forward of an equivalence class of curves.
[*] Tangent vectors as derivations
[*] The tangent space TVV of a vector space V
[*] A simple example of the push-forward operation
[*] The tangent space of a product manifold
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[*] Vector Fields and n-Forms
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[*] Vector Fields
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[*] The main definition
[*] The vector field commutator
[*] /i-related vector fields
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[*] Integral Curves and Flows
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[*] Complete vector fields
[*] One-parameter groups of diffeomorphisms
[*] Local flows
[*] Some concrete examples of integral curves and flows
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[*] Cotangent Vectors
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[*] The algebraic dual of a vector space
[*] The main definitions
[*] The pull-back of a one-form
[*] A simple example of the pull-back operation
[*] The Lie derivative
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[*] General Tensors and n-Forms
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[*] The tensor product operation
[*] The idea of an n-form
[*] The definition of the exterior derivative
[*] The local nature of the exterior derivative
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[*] DeRham Cohomology
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[*] Lie Groups
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[*] The Basic Ideas
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[*] The first definitions
[*] The orthogonal group
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[*] The Lie Algebra of a Lie Group
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[*] Left-invariant vector fields
[*] The completeness of a left-invariant vector field
[*] The exponential map
[*] The Lie algebra of GL(n,TR)
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[*] Left-Invariant Forms
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[*] The basic definitions
[*] The Cartan-Maurer form
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[*] Transformation Groups
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[*] The basic definitions
[*] Different types of group action
[*] The main theorem for transitive group actions
[*] Some important transitive actions
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[*] Infinitesimal Transformations
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[*] The induced vector field
[*] The main result
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[*] Fibre Bundles
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[*] Bundles in General
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[*] Introduction
[*] The definition of a bundle
[*] The idea of a cross-section
[*] Covering spaces and sheaves
[*] The definition of a sub-bundle
[*] Maps between bundles
[*] The pull-back operation
[*] Universal bundles
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[*] Principal Fibre Bundles
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[*] The main definition
[*] Principal bundle maps
[*] Cross-sections of a principal bundle
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[*] Associated Bundles
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[*] The main definition
[*] Associated bundle maps
[*] Restricting and extending the structure group
[*] Riemannian metrics as reductions of B(.M)
[*] Cross-sections as functions on the principle bundle
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[*] Vector Bundles
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[*] The main definitions
[*] Vector bundles as associated bundles
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[*] Connections in a Bundle
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[*] Connections in a Principal Bundle
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[*] The definition of a connection
[*] Local representatives of a connection
[*] Local gauge transformations
[*] Connections in the frame bundle
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[*] Parallel Transport
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[*] Parallel transport in a principal bundle
[*] Parallel transport in an associated bundle
[*] Covariant differentiation
[*] The curvature two-form
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[*] Bibliography
[*] Index
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