# University of California, Berkeley

Instructor:  Robert Littlejohn
Office:  449 Birge
Office Hours:   TBA
Telephone:  642-1229
Email:  physics250@wigner.berkeley.edu

Lecture:  402 LeConte
Time:  TuTh 2-3:30
Text(s):   M. Nakahara, Geometry, Topology and Physics (IOP Pubishing, Bristol,
2003), 2nd edition, and Theodore Frankel, The Geometry of Physics (Cambridge, 1997), recommended.

The textbook by Nakahara hasn't changed too much from the first (1990) edition, and you can probably get by with it
if you don't want to buy the new one.

#### Lecture Notes.

• Notes 1: Manifolds, Tangent Vectors and Covectors, in ps or pdf format, with extra figures.
• Notes 2: Differential Geometry of Lie Groups, in ps or pdf format, with extra figures.

#### Homework assignments will normally (I hope) be made available on this web site by Friday of each week, and will be due one week later, Friday at 5pm, in the envelope outside my office (449 Birge).

• Homework 1, due Friday, September 5 at 5pm in postscript or pdf format.
• Homework 2, due Friday, September 12 at 5pm in postscript or pdf format.
• Homework 3, due Friday, September 19 at 5pm in postscript or pdf format.
• Homework 4, due Friday, September 26 at 5pm in postscript or pdf format.
• Homework 5, due Friday, October 3 at 5pm in postscript or pdf format.
• Homework 6, due Friday, October 10 at 5pm in postscript or pdf format.
• Homework 7, due Friday, October 17 at 5pm in postscript or pdf format.
• Homework 8, due Friday, October 24 at 5pm in postscript or pdf format.
• Homework 9, due Friday, October 31 at 5pm in postscript or pdf format.
• Homework 10, due Friday, November 7 at 5pm in postscript or pdf format.
• Homework 11, due Friday, November 14 at 5pm in postscript or pdf format.
• Homework 12, due Friday, November 21 at 5pm in postscript or pdf format.
• Homework 13, due Friday, December 5 at 5pm in postscript or pdf format.
• Unhomework 14, nothing due, in postscript or pdf format.
• Homework 15, due Friday, December 12 at 5pm in postscript or pdf format.

#### Solutions

• Solutions for hw1.
• Solutions for hw2.
• Solutions for hw3.
• Solutions for hw4.
• Solutions for hw5.
• Solutions for hw6.

#### Course Outline

1. Physical context. Gauge theories, monopoles, instantons, order parameters, general relativity, Berry's phase, classical mechanics, etc.
2. Mathematical background. Maps, functions, hierarchy of spaces and manifolds, vector spaces, metrics, pull-backs and adjoints, raising and lowering indices, quotient spaces, group actions etc.
3. Homology and homotopy. Abelian groups, simplexes, homology groups, fundamental group, higher homotopy groups, defects in liquid crystals, textures in superfluid ${\rm He}_3$.
4. Differentiable manifolds, tangent, cotangent spaces and bundles, flows and Lie derivatives, differential forms, Stokes' theorem, differential geometry of group manifolds, Lie algebras, de Rham cohomology, Poincare' lemma, Frobenius theorem.
5. Riemannian manifolds. Parallel transport, connection, curvature, holonomy, isometries and Killing vectors, vielbeins, Hodge theory, Cartan structure equations.
6. Fiber bundles, vector bundles, principal bundles, spin and Clifford bundles, connections, holonmy, curvature, gauge field theories, general relativity as a gauge theory, geometrical phases.
7. Characteristic classes, topology of fiber bundles, Chern classes and characters, Chern-Simons forms, Stiefel-Whitney classes.

This is a little more than I covered the last time I taught the course. I may leave out some material on general relativity, which is covered in Physics 231 anyway, to make time to finish the list of topics. I may also skip some of the introductory material.