Differential Geometry (250) graduate course, Fall 2005

Monday, Wednesday, Friday 11-11.50am in HSS 2150.

Lecturer: Justin Roberts

This will be a fairly standard course in smooth manifolds and their geometry. The main subjects of the first term (250A) will be differential topology and differential forms (that is, calculus on manifolds), while the second (250B) will focus on Riemannian geometry and its applications. The third term will be taught by Ben Chow - what he covers is still to be decided.

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.)

Here are some "philosophical" links (primarily addressing the relation between geometry and physics) which you might find interesting:

THE UNREASONABLE EFFECTIVENESS OF MATHEMATICS IN THE NATURAL SCIENCES, a famous article by Eugene Wigner.

On teaching mathematics, a provocative article by Vladimir Arnold.

Geometry and Physics - A marriage made in heaven, a video lecture by Sir Michael Atiyah.

Rough syllabus: 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 Rham cohomology and examples
5. Riemannian metrics, Frobenius' theorem, symplectic and contact forms.

250B
1. Vector bundles, principal bundles
2. Connections, curvature
3. Levi-Civita connection
4. Geodesics
5. Sectional, Riemann, Ricci and scalar curvatures, connections with topology.
6. Relativity

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.

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 recommend either of John Lee's Springer books Topological manifolds or Smooth manifolds. The first one is grievously mistitled: it's actually a book on basic algebraic topology rather than the theory of topological (meaning non-smoothable) manifolds in high dimensions. The second one strikes me as going nowhere very very slowly - it's 600 pages long and covers one term's work! Having said that, my students 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 here. These are notes from Ko's USC course. 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!

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.

Additional Information

My office hours are Mon 2-3pm in APM 7210.

justin@math.ucsd.edu

"Event horizon" by Rudie Berkhout