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Spring 2011 : MATH782 Differential Geometry

  1. General information
  2. Textbook
  3. Prerequisite
  4. Homework and projects
  5. Grading scheme
  6. Course syllabus

General information

Textbook

docarmo.jpg

Manfredo Perdigao do Carmo "Riemannian Geometry", Birkhauser, 1992.

Prerequisite

The prerequisite for this class is MATH781 Differentiable Manifolds. As far as this course is concerned, the most important topics on that list are manifolds, vector bundles, vector fields, differential forms, and Lie groups. If you have not taken MATH781 but have learned these topics elsewhere you should contact the instructor.

Homework and projects

Homework assignments will be posted here every second week, next to the syllabus. After two weeks, we will go through the homework problems in class, and you will be expected to volunteer to present solutions.

There will be no midterm or final exam. Instead you will be required to work on a short project, describing a theorem or result related to Riemannian geometry. Some suggested topics will be provided. The project should consist of a written component (5 to 10 pages) to be handed in and a short (20 minute) presentation to the class toward the end of the semester.

Grading scheme

Course grades will be composed of homework (50%) and projects (50%).

Course syllabus

Topics covered will follow this rough syllabus. The more detailed syllabus below will be updated as the semester progresses.

Week

Material covered

Homework

Jan 10-14

Chapter 1 : Riemannian metrics

  • examples of metrics
  • invariant metrics on Lie groups

Homework assignment 1.
Due Friday 21st January.

Jan 17-21
No class Monday
MLK day

  • length and volume

Solutions 1.

Jan 24-28

Chapter 2 : Connections

  • affine connections
  • parallel transport
  • the Levi-Civita connection

Homework assignment 2.
Due Friday 4th February.

Jan 31-Feb 4

Chapter 3 : Geodesics

  • the geodesic flow and existence of geodesics
  • the exponential map
  • Gauss' Lemma

Solutions 2.

Feb 7-11

  • distance minimizing curves
Chapter 4 : Curvature
  • symmetries of the Riemann curvature tensor
  • sectional curvature

Homework assignment 3.
Due Friday 18th February.

Feb 14-18

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

Solutions 3.
Isometric deformation of a helicoid to a catenoid.

Feb 21-25

  • conjugate points
Chapter 6 : Isometric immersions
  • the second fundamental form
  • Gauss's formula

Homework assignment 4.
Due Friday 4th March.
Some possible topic for projects.

Feb 28-Mar 4

  • totally geodesic submanifolds
  • minimal submanifolds
Chapter 7 : Completeness
  • Hopf-Rinow Theorem

Solutions 4.
Homework assignment 5.
Due Friday 18th March.

Mar 7-11

Spring break - no classes


Mar 14-18

  • Hadamard's Theorem
Chapter 8 : Constant curvature
  • Cartan's Theorem

Solutions 5.

Mar 21-25

  • models for hyperbolic space
  • space forms
  • Bieberbach's Theorem

Homework assignment 6.
Due Friday 1st April.

Mar 28-Apr 1

  • isometries of hyperbolic space
Chapter 9 : Variations of energy
  • first variation of energy

Solutions 6.

Apr 4-8

  • second variation of energy
  • Bonnet-Myers Theorem
  • Synge-Weinstein Theorem

Homework assignment 7.
Due Friday 15th April.

Apr 11-15

Symmetric spaces

  • completeness
  • locally symmetric spaces
  • structure of the Lie algebra of Killing fields

A reference for symmetric spaces is Section 5.3 of Riemannian geometry and geometric analysis by Jurgen Jost.
Solutions 7.

Apr 18-22
No class Friday
Good Friday

  • curvature
Wednesday presentations: Dan, Merrick, Nate

Apr 25-29
Last class on Wednesday

Monday presentations: Bevan, John, Mayukh
Wednesday presentations: Adam, Andrea, David, Elise, Justin, Michael

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