Julius Ross

Part III Differential Geometry 2011

I am lecturing the foundations of differential geometry, including manifolds, vector bundles, connections and curvature. Lectures MWF 9am MR9 (Note: This course was originally to be lectured by Prof. Wilson)

Prerequisites

The course will not assume and familiarity with differential geometry, but will take a rather swift (and narrow) course through the fundamentals. Prerequsities include multivariable differentiation (as in Cambridge Analysis II) and familiarity with basic topological spaces (i.e. the notion of open set and open cover).

Sources

I would most recommend the following two texts:

• Introduction to Smooth Manifods (Lee. GTM 218)
• Riemannian manifolds: an introduction to curvature (Lee. GTM 176). Excellent for the part of the course on connections and curvature.

There are many other good sources including the following selection

• A Comprehensive Introduction to Differential Geometry (Michael Spivak). A gentle but massive five volume series, but only volume 1 is relevant.
• Calculus on Manifolds (Michael Spivak). A book for the required multivariable analysis
• Foundations of Differentiable Manifolds and Lie Groups (Warner. GTM 94)
• Will Merry wrote up some lecture notes from a previous Cambridge course which can be found on Prof. Wilson's website. The current course will be similar, but not identical.
• Lectures on Lie Groups (Adams) amazon
• Nigel Hitchin has some notes online which I highly recommend.
• A large number of detailed handouts can be found on Brian Conrad's webpage.

Handouts
Existence of partitions of unity (This will only be examinable under the additional assumption that the manifold is compact)
Boundary orientation This handout fixes a silly mistake I made with regard to the definition of the orientation on the boundary
Existence of the Levi-Civita connection

Exercise Sheets
Exercise Sheet 1 (19 October 2011, version 2 with one added exercise)
Exercise Sheet 2 (5 December 2011, version 2 with correction to question 3 and 8)
Exercise Sheet 3 (28 November 2011)

Here is a summary of the topics that were covered in the course in 2011.

Mathematical Tripos

The dates for exams the mathematical tripos can be found here.

Misc Material

Proof of the Chain Rule

Monkey Saddle

Not much else to see, but you are welcome to look at material from my past courses.

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