MA4CO Differential Geometry, Spring 2016

Administrative Matters

  • Lectures
    1. Monday, 17:00-18:00, MS.05
    2. Thursday, 15:00-16:00, MA_B3.01
    3. Friday, 11:00-12:00, MA_B3.02
  • Examples Class
    • TA: Louis Bonthrone
    • Time: TBA
    • Location: TBA
  • Contact
  • Assessment
    • 100% Final examination.

Recommended Reading

Text Books

  • Do Carmo: Riemannian Geometry (a classic text that is certainly relevant today but sometimes considered a little terse)
  • Lee: Riemannian Manifolds: An Introduction to Curvature (very readable, possibly a little elementary in places)
  • Chavel: Riemannian Geometry: A Modern Introduction (more advanced, extensive discussion of many aspects of Riemannian Geometry)
  • Petersen: Riemannian Geometry (more advanced, slightly non-standard approach definitely worth a look at some point)
  • Gallot, Hulin, Lafontaine: Riemannian Geometry (more advanced, but very nice development of the formalism of Riemannian Geometry)

Lecture Notes

Differentiable Manifolds

Some sources for differential manifolds. There are many resources available, and some of the resources listed above treat this topic before moving on to Riemannian Geometry. The following should be sufficient background reading.

Course Summary

This course is about Riemannian geometry, that is the extension of geometry to spaces where differential/integral calculus is possible, namely to manifolds. We will study how to define the notions of length, angle and area on a smooth manifold, which leads to the definition of a Riemannian Manifold. Important concepts are

  • Riemannian metrics
  • Connections (differentiation of vector fields)
  • Length, angle
  • Geodesics (shortest paths)
  • Area
  • Curvature

At the end of the course, we will touch on some global aspects of Riemannian geometry, and discuss a little about the interaction between curvature, geometry and topology. Such interaction was studied heavily in the mid to late 20'th century and is currently still an active area of research. A famous example is the Hamilton-Perelman resolution of the Poincaré conjecture, one of the Clay Foundation's seven Millennium Prizes, was resolved only this century.

Lecture Diary

Week 1

Below you will find brief notes on each lecture, sketching the material covered in that lecture. I will try to provide references to the corresponding chapters in the reading material.

  • Lecture 1: Course Introduction
  • Lecture 2: Review Of Smooth Manifolds
  • Lecture 3: Riemannian Manifolds