Mathematics – Logic
Scientific paper
Jun 2006
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006cqgra..23.4543i&link_type=abstract
Classical and Quantum Gravity, Volume 23, Issue 13, pp. 4543-4544 (2006).
Mathematics
Logic
Scientific paper
Most of us sometimes have to face a student asking: 'What do I need to get started on this'. (In my case 'this' would typically be a topic in general relativity.) After thinking about it for quite a while, and consulting candidate texts again and again, a few days later I usually end up saying: read this chapter in book I (but without going too much detail), then that chapter in book II (but ignore all those comments), then the first few sections of this review paper (but do not try to work out equations NN to NNN), and then come back to see me. In the unlikely event that the student comes back without changing the topic, there follows quite a bit of explaining on a blackboard over the following weeks.
From now on I will say: get acquainted with the material covered by this book. As far as Isham's book is concerned, 'this' in the student's question above can stand for any topic in theoretical physics which touches upon differential geometry (and I can only think of very few which do not).
Said plainly: this book contains most of the introductory material necessary to get started in general relativity, or those branches of mathematical physics which require differential geometry. A student who has mastered the notions presented in the book will have a solid basis to continue into specialized topics. I am not aware of any other book which would be as useful as this one in terms of the spectrum of topics covered, stopping at the right place to get sufficient introductory insight.
According to the publisher, these lecture notes are the content of an introductory course on differential geometry which is taken by first-year theoretical physics PhD students, or by students attending the one-year MSc course 'Quantum Fields and Fundamental Forces' at Imperial College, London. The volume is divided into six chapters: An Introduction to Topology Differential Manifolds Vector Fields and n-Forms Lie Groups Fibre Bundles Connections in a Bundle.
It is a sad reflection on current academic curriculi that Chapter I is needed at all. This is in fact the chapter that I liked least in the book.
The presentation has the right balance between formal definitions and introductory comments to make the book accessible for self-study. Most definitions are followed by excellent examples, though this is not uniform: more examples would have been helpful in several places, both in case of self-study and to make life easier for someone lecturing from this book. The most notable lacuna is (essentially) the lack of exercises: surely there would have been many worked out during the course at Imperial, and it is a pity that they have not been included in the book. I very much hope that there will be a further edition with lots of examples and exercises (note that the latter can also play the former role), making this work even more useful.
The first chapter of the book is a crash course on topology, covering metric spaces, orders, lattices, convergence, compactness, as well as separation axioms. The introduction of filters might be seen as unnecessarily advanced, in view of a few notable gaps: the first of those concerns connectedness, which lies at the heart of many proofs, and which is only mentioned in a footnote on page 61; it deserves a small subsection of its own. The second gap is paracompactness, related to existence of partitions of unity, which is a key to several constructions on manifolds; again the notion only appears as a footnote on page 231. A short discussion of the Kuratowski Zorn lemma might have been useful. Fortunately, the students that are likely to come to my office will already be familiar with the material in this chapter (and more, as far as topology is concerned), so this is not really an issue from my point of view. On the other hand, it could be one for somebody lecturing from this book.
In the two following chapters the notion of a manifold is introduced, with a careful discussion of tangent space, vector fields and their flows. Covectors, exterior differentiation, and tensors are introduced. This brings me again to the notion of paracompactness: some authors choose to add the requirement of paracompactness to the definition of a manifold, and I am very much in favour of such an approach, as then various pathologies are avoided. The fact that this has not been done cannot be seen as a criticism of this book, as several other textbooks do not make this assumption, but this would be my suggestion to anyone lecturing on the topic. I did not like the notation A for exterior algebra (why not use Λ like many authors?).
Chapter 4 constitutes an excellent introduction to Lie groups, and algebras. This is my favourite chapter in the book. This subject is rarely presented at an elementary level, and I highly recommend the book to anyone looking for a concise introduction. (As a very minor point, I did not like the notation [AB] for the commutator (what's wrong with [A,B]?)
Chapter 5 discusses fibre bundles. I did not like the definition of a fibre bundle which does not assume local triviality, with all fibres modelled on one single space. This extension of the notion might be useful in some applications, but it is certainly not standard. I am strongly against using non-standard definitions in introductory texts, as this leads to confusions and misunderstandings.
Definitions are of course a matter of convention, but they provide a means of communication, and communication is broken if one starts changing those definitions arbitrarily. Apart from that, this is again a useful introduction to various bundles, including principal ones, and those associated to the latter. Chapter 6 is a logical continuation into the world of connections, and parallel transport.
I hope to have made it clear that my critical remarks are secondary, and that this is a very useful and readable book overall, a copy of which (or more) should be on the shelves of the library of any institution with graduate students in mathematics or physics. I would be delighted to see a new, extended, edition with the definition of fibre bundles streamlined, and more examples included.
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