Justin Roberts
This will be a fairly standard course in smooth manifolds, covering
primarily differential topology and differential forms (that is, calculus
on manifolds). Next term it will mutate into Lie group theory 251BC which
is a shame from my point of view because there's so much beautiful maths
to explain!
This webpage was originally written for a full-year course called
"Geometry and Physics" (259ABC) which I taught essentially as
"Differential Geometry with special reference to physical motivations and
applications", but I haven't bothered removing any of the extra stuff
because it's probably still helpful for some people.
There are several different paths one can take through differential
geometry. My interest in it is mainly in how it relates to topology,
physics, and gauge theory, so my course will certainly mention some of
these applications. My main influences have been people like Atiyah, Bott,
Segal and Milnor, rather than people studying Riemannian geometry, so I
try to emphasise conceptual geometric ideas and underlying algebraic
structures more than analysis and local calculations (I will try to keep
outbreaks of indices to a minimum.)
Pre-requisites. This course is not a qualifying course; the great thing
about that is that we don't have to have exams, so can concentrate on
trying to learn and understand instead! It is probably normally thought of
as a "second-year" graduate course. However,I want it to be accessible
also to first years and to physicists, so I am not going to insist on any
previous knowledge of geometry or topology. I can't promise that the
course will be entirely self-contained, but I will try to make sure that I
give at least a rough explanation of anything "external" I need, and
encourage people to ask about or read up on things they aren't familiar
with.
Rough syllabus for 250A:
1. Manifolds, smooth functions, smooth manifolds
2. Tangent bundle, derivatives, basic differential topology
3. Vector fields and flows
4. Differential forms, integration, Stokes' theorem, de Rhamcohomology
and examples
5. Riemannian metrics, Frobenius' theorem, symplectic and contactforms.
(If there was a 250B, it would deal with the following sorts of
thing:
1. Vector bundles, principal bundles
2. Connections, curvature
3. Levi-Civita connection
4. Geodesics
5. Sectional, Riemann, Ricci and scalar curvatures, connections
withtopology.
6. Relativity.)
Books. I don't intend to follow any single book too closely, but
here are some of the most obvious references:
A. Differential Topology and Forms
Topology from the differentiable viewpoint, by John Milnor
(Princeton). This is a very short and very nice account of the basics of
smooth manifold theory and differential topology. Milnor is God.
Differential Topology by Moe Hirsch (Springer). This is a fuller
account of differential topology, with a slightly more analytical flavour
(he states all the theorems for r-times differentiable functions rather
than just for smooth ones) than Milnor's book. It's also very good.
Differential forms in algebraic topology, by Bott and Tu
(Springer). This book covers the area between differential topology and
algebraic topology. It isn't exactly suitable for either of those courses
separately, because of its unusual structure, but anyone who wants to
really get to grips with the combination should read it. It's beautifully
written and full of illuminating examples.
An introduction to differential manifolds, by Dennis Barden and
Charles Thomas (Imperial College Press). My old Cambridge teachers have
published their lecture notes on differential forms, and very nice they
are too. The book doesn't go very far, but it is very thorough in what it
covers, namely, everything you need to understand de Rham cohomology.
By the way, I don't really recommend either of John Lee's Springer
books Topological manifolds or Smooth manifolds. The first
one mostly because it's annoyingly mistitled: it's actually a book on
basic algebraic topology rather than the theory of topological (meaning
non-smoothable) manifolds in high dimensions (which is a subject where
there actually need to be more books!). The second one strikes me as
just too slow - it's 600 pages long and covers one term's work! Having
said that, my students have generally found it very readable.
B. Riemannian geometry
Morse theory, by John Milnor (Princeton). This is an absolute
classic, perhaps the greatest set of mathematical lecture notes ever.
Milnor packs a scarcely-believable amount of beautifully-explained
geometry and topology into just 120 pages. All subsequent books on
differential geometry appear to be based on this one.
Riemannian geometry, by Guillot, Hulot and Lafontaine (Springer).
This is a very nice book; it gives a lot of geometric intuition as well as
a clear presentation. The only drawback from my point of view is its
minimisation of the use of differential forms.
Semi-Riemannian geometry, by Barrett O'Neill (Academic Press). This
is a good mathematical treatment of Riemannian geometry and general
relativity.
Riemannian geometry, by Peter Petersen (Springer). A nice modern
book for hardcore Riemannian geometers. Goes further in the analytic
direction than all the others here.
Riemannian Geometry, by Manfredo Do Carmo (Birkhauser). This is a
fleshed-out version of Milnor's book. Students seem to hate this for some
reason.
C: Other useful things
Notes on differential geometry, by Ko Honda, available on his
webpage at UCLA. They are very strongly modelled (I assume) on Raoul
Bott's Harvard lectures, so the pedigree is impeccable, and in my view
they display really excellent taste and choice of illuminating examples,
especially at the beginning. They don't go very much into Riemannian
geometry, but are very good on differential forms and the theory of
connections (foundations of gauge theory).
The geometry of physics, by Ted Frankel (Cambridge). This book has
a lot of very nice presentations of geometrical and topological ideas in
physics. There are umpteen books of this kind (i.e. geometry for
physicists), but for my money Ted's is the most mathematically sound. What
we really need now is a book of physics for geometers!
Gauge Fields, Knots and Gravity
by John Baez and Javier Muniain (World Scientific). This is a great book
which I'd forgotten about until recently. John Baez is a great expositor
of both maths and physics, and his approach is very much in the spirit I
like. He takes care to explain many of the useful, beautiful and/or
irritating things which most books tend to ignore and tries
consistently to extract meaningful interpretations from formulae (e.g. "what does the Bianchi
identity actually mean geometrically?").
Mathematical methods of classical mechanics, by Vladimir Arnold
(Springer). Arnold's book may not appear directly relevant, but his
completely geometric presentations of many concepts (especially his
appendix B on Riemannian geometry) are wonderfully clear and sadly rare in
mathematical literature. This is a classic and brilliant book by a
champion of the organic "Newtonian" (otherwise known as "Russian")
approach to mathematics - and a sworn enemy of tedious Bourbakiste
abstraction.
Topology and Geometry, by Glen Bredon (Springer). This book tries
to teach algebraic topology from the point of view of smooth manifolds,
which means that it begins with differential forms. Though it isn't
anywhere near as beautiful as its model, Bott and Tu, it is more
comprehensive, and looks OK to me. Students who've used it all seem to
hate it though.
Links. Here are some
"philosophical" links (primarily addressing the relation between geometry
and physics) which you might find interesting:
