MATH 6250I - Riemannian Geometry

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Syllabus [PDF]

Course Description

An introductory course on Riemannian Geometry targeted at:

  • postgraduate students in mathematics (both pure and applied);
  • advanced undergraduate students who are strongly interested in geometry and topology;
  • physics students who need background knowledge for studying general relativity.

Tentative Outline

  • Geometry of Euclidean Hypersurfaces
  • Riemannian Metrics and Connections
  • Geodesics and Parallel Transport
  • Curvatures of Riemannian Manifolds
  • Jacobi Fields and Index Forms
  • Introduction to Ricci Flow

Prerequisites

Good understanding of differentiable manifolds and tensor calculus (MATH 4033 or 5230). For 4033, Chapters 2 and 3 are the most important -- you need to understand what is a manifold, what are tangent and cotangent spaces, tangent maps, tensors and differential forms, etc.

Reference Books

Free books (PDFs available within campus network)

  • Riemannian Manifolds by John M. Lee [here]
  • Riemannian Geometry by S. Gallot, D. Hulin, J. Lafontaine [here]
  • Riemannian Geometry by Peter Petersen [here]
  • Riemannian Geometry and Geometric Analysis by Jürgen Jost [here]

Pay books

  • Hamilton's Ricci Flow (Chapter 1) by B. Chow, P. Lu, L. Ni
  • Riemannian Geometry by M.P. Do Carmo
  • A Comprehensive Introduction to Differential Geometry, Vol. 1 by Michael Spivak
  • Manifolds and Differential Geometry by Jeffrey M. Lee

Lecture Notes

Complete Set (including 4033 portion): [Download]

We will use the following journal paper for reference in the Ricci Flow chapter:

Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255--306 [Link]

Homework

Problem Set 1 [Downlaod][Selected Solutions*] (Due: 30/09/2018)
Problem Set 2 [Download] (Due: 04/11/2018)
Problem Set 3 [Download] (Due: 09/12/2018)

* Only solutions to selected problems with simpler approaches than most students' submissions.

Submit either

  • a neatly written, scanned PDF (1 combined file); or
  • a neatly written PDF exported from Tablet apps (e.g. GoodNotes, Notability, etc.); or
  • a LaTeX-typed PDF

to Canvas before the deadline.

Right after the deadline, your homework will be randomly circulated to three classmates for peer review, and you will also receive three others' homework for review.

Whiteboard Slides

[Week 1][Week 2][Week 3][Week 4][Week 5][Week 6][Week 7][Week 8][Week 9][Week 10][Week 11][Week 12][Week 13]

Presentations

According to the grading policy in the syllabus, you can give an optional presentation on a journal paper if you hope to be considered for an A+ grade. You will be given a medium length (~15 pages) to read. In the presentation you should outline the paper, and go through one proof (selected by the instructor about 1 month in advance) in detail.

Click [here] to see the list of papers. Discuss with the instructor (in person or through Email) on which paper, and which part of it you may do the presentation.

Links to student presentations: [here]

 

Course Summary:

Date Details Due