Modern Differential Geometry for Physicists by Isham

[micromass]

• Author: C.J. Isham
• Title: Modern Differential Geometry for Physicists
• Amazon Link: https://www.amazon.com/dp/9810235623/?tag=pfamazon01-20

Table of Contents:

[LIST]
[*] An Introduction to Topology
[LIST]
[*] Preliminary Remarks
[LIST] 
[*] Remarks on differential geometry
[*] Remarks on topology
[/LIST]
[*] Metric Spaces
[LIST]
[*] The simple idea of convergence
[*] The idea of a metric space
[*] Examples of metric spaces
[*] Operations on metrics
[*] Some topological concepts in metric spaces
[/LIST]
[*] Partially Ordered Sets and Lattices
[LIST]
[*] Partially ordered sets
[*] Lattices
[/LIST]
[*] General Topology
[LIST]
[*] 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
[/LIST]
[/LIST]
[*] Differentiable Manifolds
[LIST]
[*] Preliminary Remarks
[*] The Main Definitions
[LIST]
[*] Coordinate charts
[*] Some examples of differentiable manifolds
[*] Differentiable maps
[/LIST]
[*] Tangent Spaces
[LIST]
[*] 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
[/LIST]
[/LIST]
[*] Vector Fields and n-Forms
[LIST]
[*] Vector Fields
[LIST]
[*] The main definition
[*] The vector field commutator
[*] /i-related vector fields
[/LIST]
[*] Integral Curves and Flows
[LIST]
[*] Complete vector fields
[*] One-parameter groups of diffeomorphisms
[*] Local flows
[*] Some concrete examples of integral curves and flows
[/LIST]
[*] Cotangent Vectors
[LIST]
[*] 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
[/LIST]
[*] General Tensors and n-Forms
[LIST]
[*] The tensor product operation
[*] The idea of an n-form
[*] The definition of the exterior derivative
[*] The local nature of the exterior derivative
[/LIST]
[*] DeRham Cohomology
[/LIST]
[*] Lie Groups
[LIST]
[*] The Basic Ideas
[LIST]
[*] The first definitions
[*] The orthogonal group
[/LIST]
[*] The Lie Algebra of a Lie Group
[LIST]
[*] Left-invariant vector fields
[*] The completeness of a left-invariant vector field 
[*] The exponential map
[*] The Lie algebra of GL(n,TR)
[/LIST]
[*] Left-Invariant Forms
[LIST]
[*] The basic definitions
[*] The Cartan-Maurer form
[/LIST]
[*] Transformation Groups
[LIST]
[*] The basic definitions
[*] Different types of group action
[*] The main theorem for transitive group actions
[*] Some important transitive actions
[/LIST]
[*] Infinitesimal Transformations
[LIST]
[*] The induced vector field
[*] The main result
[/LIST]
[/LIST]
[*] Fibre Bundles
[LIST]
[*] Bundles in General
[LIST]
[*] 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
[/LIST]
[*] Principal Fibre Bundles
[LIST]
[*] The main definition
[*] Principal bundle maps
[*] Cross-sections of a principal bundle
[/LIST]
[*] Associated Bundles
[LIST]
[*] 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
[/LIST]
[*] Vector Bundles
[LIST]
[*] The main definitions
[*] Vector bundles as associated bundles
[/LIST]
[/LIST]
[*] Connections in a Bundle
[LIST]
[*] Connections in a Principal Bundle
[LIST]
[*] The definition of a connection
[*] Local representatives of a connection
[*] Local gauge transformations
[*] Connections in the frame bundle
[/LIST]
[*] Parallel Transport
[LIST]
[*] Parallel transport in a principal bundle
[*] Parallel transport in an associated bundle
[*] Covariant differentiation
[*] The curvature two-form
[/LIST]
[/LIST]
[*] Bibliography
[*] Index 
[/LIST]

 

[Fredrik]

I think this is a very nice book for physics students who want to begin to understand the mathematics of gauge theories. The early chapters cover the basics of manifolds. I especially enjoyed the presentation of tangent spaces. Then he moves on to stuff like Lie group actions and fiber bundles. He doesn't cover things like the Riemann curvature tensor, which is used in general relativity. So it's clearly meant as preparation for gauge theories, not general relativity.

The level of rigor is below that of a typical math book. For example if he's supposed to prove that a function is a Lie group homomorhism, he proves that it's a group homomorphism and doesn't even bother to check if the function is continuous. (I don't remember if he did that exact thing. It's been a while since I read it).

The biggest problem with the book is that it ends too soon. It doesn't actually get to the point where you can write down the Yang-Mills action and obtain the field equations from it.

 

[mathwonk]

i don't know this book but my mouth dropped at the remark it does not cover the curvature tensor, given the title. I am not an expert, but I always thought that curvature was the central idea in differential geometry. does it cover it in some other more intuitive way? or do some physicists perhaps consider "modern differential geometry" a catch phrase for the general language of manifold theory?

Of course Spivak's Differential Geometry vol. 1, also treats mainly manifolds and bundles, but he makes clear in his introduction that this is only the modern language of differential geometry and not the substance, which begins in his vol.2 on curvature.

well it does look as if the last chapter of Isham's book is about connections, a geometric topic leading to curvature, and that the "curvature 2-form" occurs in the absolute last paragraph of the book, which amazon will not let me read.

 

[Fredrik]

I agree that the title is a bit misleading. I don't think this reflects what physicists mean by "modern differential geometry". I think the word "modern" might be referring to the fact that tensor fields are defined as sections of bundles, instead of by the old (and horrible) "something that transforms like this" definition.

I think "Modern differential geometry for theoretical particle physicists" would have been an accurate title. Physicists who work with general relativity probably wouldn't say that the book is for them.

No, the book doesn't cover curvature in a more intuitive way. No part of the book tries to explain curvature. It does however end with the following comment:

This completes the derivation of the basic relation between the mathematical theory of connections in principal and associated bundles, and the physicists' familiar theory of the Yang-Mills field and its gauge transformations. Note that, although we have talked above about the Yang-Mills field, the same analysis applies also to the Riemannian connection in the $GL(m,\mathbb R)$-bundle of frames $\mathbf{B}(\mathcal M)$. In this case, the parallel transport and covariant derivatives coincide with the familiar operations from elementary Riemannian geometry, and the curvature 2-form that takes its values in the Lie algebra of $GL(m,\mathbb R)$ is nothing but the usual curvature tensor in a non-holonomic basis.

It's a little odd to talk about "the usual curvature tensor" there, since the book starts with the most basic stuff and doesn't contain a section on curvature.

 

[mathwonk]

well a quick comparison of the topics in the last chapter (6) of Isham, with vol.2 of Spivak, shows that Isham devotes less than 25 pages to the last, most abstract, version of "curvature" that Spivak discusses in his last chapter (8), vol.2, in the guise of a lie algebra valued "curvature 2-form", in the context of connections on the bundle of frames. This occurs in Spivak after spending over 300 pages explaining what it has to do with actual curvature, in more elementary, but increasingly sophisticated language.

I am led to recall a sentence in Spivak's introduction to vol.1, which says roughly: if you want to write an introduction which exposes the geometry of the subject, there is no point introducing the "curvature tensor" without explaining where it came from and what it has to do with curvature. [But opinions may differ.]

In any case this is why it helps us to have Fredrik's summary of what really is in the book at hand.

If one wants a quick survey of what a mathematician thinks of as modern differential geometry, there is "part 2" of Milnor's pamphlet "Morse theory", called a rapid course in Riemannian geometry, where he summarizes connections, covariant differentiation, parallel transport, and curvature in 12 pages, with many proofs. In short, once one has covariant derivatives, "curvature" is a measure of the failure of repeated covariant differentiation to be symmetric in the two differentiation directions. This is not quite as abstract as Isham's version in his chapter 6, (not involving bundles of frames).