Fall 2009 : MATH676 (section 1) Topics in Riemannian Geometry

  1. General information
  2. Prerequisites
  3. Textbook
  4. Homework
  5. Grading scheme
  6. Course syllabus

General information

Prerequisties

For this course some knowledge of differential topology will be extremely helpful. While it would be useful to have taken MATH670 Introduction to Differential Topology (or an equivalent class), you really only need some basic familiarity with manifolds, tangent vectors, the tangent bundle, and vector bundles. I will also review some of this material in class, so you should find this class manageable even if you haven't studied differential topology before. Some knowledge of multivariable calculus, linear algebra, and elementary group theory (all at the undergraduate level) will also be assumed.

Textbook

I mainly plan to stick to do Carmo's "Riemannian Geometry" and would encourage you to purchase a copy from the bookstore (it should cost less than $50). It may be difficult to fully appreciate until you've looked at other books on Riemannian geometry, but the subject is really blessed by the existence of do Carmo's book!

docarmo.jpeg

Manfredo Perdigao do Carmo "Riemannian Geometry", Birkhauser Boston.

Homework

I plan to post a couple of homework problems each week (next to the syllabus). I expect you to attempt all the problems. However, you won't need to hand in solutions. Rather, you should take it in turns to present solutions to these problems during Wednesday's class. You are more than welcome to discuss these problems with me during office hours; if it is your turn to present, I will give you all the help you need to solve the problem beforehand.

Grading scheme

Course grades will be determined based on attendance and participation. By "participation" I mean presenting homework problems as described above.

Course syllabus

The syllabus below will be updated as the semester progresses.

Week

Material covered

Homework

Aug 24-28

Overview of Riemannian geometry
Chapter 0 : Differentiable manifolds

  • manifolds, tangent vectors
  • tensors and differential forms

No homework yet.

Aug 31-Sep 4

  • bracket on vector fields
  • Lie derivative
Chapter 1 : Riemannian metrics
  • left-invariant metrics on Lie groups

Exercise sheet 1.
Due Wednesday 9th September.

Sep 7-11
No class Monday
(Labour Day)

Chapter 2 : Connections

  • affine connections
  • the Levi-Civita connection

Exercise sheet 2.
Due Wednesday 16th September.

Sep 14-18

Chapter 3 : Geodesics

  • the geodesic flow
  • the exponential map
  • Gauss's Lemma

Exercise sheet 3.
Due Wednesday 23rd September.

Sep 21-25

  • length minimizing curves
  • geodesics in Lie groups

Continue with previous exercises.

Sep 28-Oct 2

Chapter 4 : Curvature

  • the Riemann curvature tensor
  • sectional curvature

Exercise sheet 4.
Due Wednesday 7th October.

Oct 5-9

  • Ricci and scalar curvature
Chapter 5 : Jacobi fields
  • the Jacobi equation

Exercise sheet 5.
Due Wednesday 14th October.

Oct 12-16

  • conjugate points
Chapter 7 : Hopf-Rinow and Hadamard Theorems
  • geodesic and metric completeness
  • non-positively curved manifolds

Exercise sheet 6.
Due Wednesday 21st October.

Oct 19-23

Chapter 8 : Spaces of constant curvature

  • Cartan's Theorem
  • models for hyperbolic space
  • Hilbert's Theorem

Exercise sheet 7.
Due Wednesday 28th October.

Oct 26-30

  • space forms
  • constant curvature K=1 (S^n, RP^n, Lens spaces)
  • flat manifolds (tori and their quotients)
  • negative curvature (Cartan and Preissman's Thms)

No homework (snow week).

Nov 2-6

  • Liouville's Theorem
Chapter 9 : Variations of energy
  • formula for the 1st variation of energy
  • formula for the 2nd variation of energy

Exercise sheet 8.
Due Wednesday 11th November.

Nov 9-13

  • Bonnet-Myers's Theorem
  • Weinstein's and Synge's Theorems
Chapter 6 : Isometric immersions
  • the second fundamental form

Exercise sheet 9.
Due Wednesday 18th November.

Nov 16-20

  • totally geodesic submanifolds
  • minimal submanifolds

Exercise sheet 10.
Due Wednesday 2nd December.

Nov 23-27
No classes

Thanksgiving break

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This page last modified by Justin Sawon
Friday, 20-Nov-2009 15:40:30 MST
Email corrections and comments to sawon@math.colostate.edu