Modern Geometry: Mathematics GR6402 (Fall
Tuesday and Thursday
This is the first part of a full-year course on differential
geometry, aimed at first-year graduate students in mathematics,
while also being of use to physicists and others. The spring
semester second part of the course will be taught by Simon Brendle,
and concentrate on topics in Riemannian geometry.
An undergraduate course of some kind dealing with differentiable
manifolds, such as our GU4081, Introduction to Differentiable
Manifolds, will be assumed. Much of the material of such a
course will be reviewed quickly during the early part of this
1. Review of the definition of a differentiable manifold, tangent
spaces, vector fields.
2. Tensor algebra, tensor fields, differential forms.
3. de Rham cohomology.
4. Lie groups and Lie algebras.
5. Principal bundles, vector bundles.
6. Connections and curvature on a principal bundle. Connections and
curvature for vector bundles.
7. Chern-Weil theory.
8. Frame bundles, torsion, metrics.
9. Laplacians, Hodge theory.
10. Gauge theory, Maxwell's equations, Yang-Mills equations.
11. Clifford algebras, spinors, the Dirac equation.
Tuesday, September 5: Introduction to course. Definition
of a differentiable manifold.
Thursday, September 7: Examples of differentiable
manifolds. The tangent space.
Tuesday, September 12: More on the tangent space. The
differential of a smooth map. Vector fields. The Lie bracket.
Definition of a Lie algebra.
Thursday, September 14: The tangent bundle. Vector bundles and
sections. The cotangent bundle. Differential 1-forms.
Tuesday, September 19: Tensor products. Tensor fields.
Thursday, September 21: Tensors as multi-linear
functions. Symmetric and exterior algebras. Differential
Tuesday, September 26: The exterior derivative and its
Thursday, September 28: Some symplectic geometry and
Hamiltonian mechanics. The Lie derivative.
Tuesday, October 3: Lie derivative of vector fields. de
Rham cohomology. Poincaré lemma.
Thursday, October 5: Some homological algebra. Mayer-Vietoris
sequence, computation of cohomology of spheres. Comments on de Rham
Tuesday, October 10: Orientations, integration of differential
forms, Stokes theorem.
Thursday, October 12: Lie groups: definitions and examples.
Tuesday, October 17: Submanifolds. Actions of Lie groups on
Thursday, October 19: Lie algebras: left-invariant vector
fields, the exponential map.
Tuesday, October 24: The adjoint representation.
Baker-Campbell-Hausdorff formula. Left-invariant differential forms
on Lie groups.
Thursday, October 26: The Maurer-Cartan equation. Lie algebra
Tuesday, October 31: Principal bundles, the gauge group.
Thursday, November 2: Connections on principal bundles.
Curvature. Bianchi identities.
Thursday, November 9: Frame bundles, torsion. Cartan
Tuesday, November 14: Riemannian geometry, the Levi-Civita
connection. Formulas in local coordinates.
Thursday, November 16: More about local coordinates.
Topology of bundles, examples.
Tuesday, November 21: Chern-Weil theory.
Tuesday, November 28: More Chern-Weil theory, Maxwell's
Thursday, November 30: Yang-Mills equations, General
Tuesday, December 5: Covariant derivatives and connections on
Thursday, December 7: Clifford algebras, spinors and the Dirac
Thursday, December 21: Final Exam 9-12 am Math 307
1 (due Thursday, September 14)
2 (due Thursday, September 21)
3 (due Thursday, September 28)
4 (due Thursday, October 5)
5 (due Thursday, October 12)
6 (due Thursday, October 19)
7 (due Thursday, October 26)
8 (due Thursday, November 2)
9 (due Thursday, November 16)
10 (due Thursday, November 30)
11 (due Thursday, December 7)
12 (due Thursday, December 21)
There will be an in-class final exam 9-12 am Thursday December 21.
I should be available after class in my office (Math 421).
Feel free to come by Math 421 at any time and I will likely have
some time to talk, or make an appointment by emailing me.
The teaching assistant for the course is Pei-ken Hung
(email@example.com). He will be available for help with the
material of the course Fridays 9-12.
Books and Online Resources
Some recommended textbooks:
Morita, Geometry of Differential Forms.
Kobayashi and Nomizu, Foundations of Differential Geometry,
Spivak, A Comprehensive Introduction to Differential Geometry,
Volumes 1 and 2.
of Differentiable Manifolds and Lie Groups.
Introduction to Manifolds (this is an
undergraduate level book, we will assume much of this material).
Geometry: Connections, Curvature and Characteristic Classes.
Lee, John, Introduction
to Smooth Manifolds.
Lee, Jeffrey, Manifolds and Differential Geometry.
Bleecker, David, Gauge Theory and Variational Principles.
Sharpe, Differential Geometry: Cartan's Generalization of
Klein's Erlangen Program.
Geometry: Bundles, Connections, Metrics and Curvature.
Baum, Helga, Eichfeldtheorie.
(only available in German).
An excellent review article outlining the subject and the relation
Eguchi, Gilkey and Hanson, Gravitation,
Gauge Theories and Differential Geometry.
An excellent series of lecture notes on connections on bundles and
Figueroa-O'Farrill, Gauge Theory