Jan 8 2022 · 2 min read

This is the page for the Riemannian Geometry reading group I’m organizing for the winter of 2022. The idea is to explore some basic concepts first to get everyone up to speed, then branch out into more material later on. Below is a list of material along with the week number in which I’d like to get to it, though just the first few weeks for now. I’ll be updating it as we go along.

Week | Content | Reference 1 | Reference 2 | Reference 3 |
---|---|---|---|---|

1 | Tensors, Review of Diff Geo | LeeRM Ch 2-3 | Warner Ch 2, G&Q1 Ch14 | Kb&N1 Ch 1.1-1.3 Ch 4.1 |

2 | Metrics, Covariant Derivatives / Connections, Geodesics | LeeRM Ch 3-4 | G&Q1 Ch 15 | |

2.5 | Geodesics, Riemannian Distances, Hopf-Rinow Theorem | LeeRM Ch 5-6 | G&Q1 16 | Kb&N1 3.6, 4.1-4.5 |

3 | Curvature and beyond | LeeRM Ch 7-8 | G&Q1 Chapter 17-18 |

Notes

- We are using the first edition of John Lee’s Riemannian Manifolds (link has been updated!)
- For week 2, if you are looking for a review of differential forms and cohomology, see chapters 14,17 in LeeSM

Along with a legend of books:

Short Title | Full Title | Authors |
---|---|---|

LeeRM | Introduction to Riemannian Manifolds 2nd Edition | John Lee |

LeeSM | Introduction to Smooth Manifolds 3rd Edition | John Lee |

G&Q1 | Differential Geometry and Lie Groups: A first course | Jean Gallier, Jocelyn Quaintance |

G&Q2 | Differential Geometry and Lie Groups: A second course | Jean Gallier, Jocelyn Quaintance |

Warner | Foundations of Differentiable Manifolds and Lie Groups | Frank Warner |

Kb&N1 | Foundations of Differential Geometry Vol 1 | Shoshichi Kobayashi, Katsumi Nomizu |

Kb&N2 | Foundations of Differential Geometry Vol 2 | Shoshichi Kobayashi, Katsumi Nomizu |

Some other cool stuff I’d like to cover eventually:

- Hodge Theory : Gives a duality theory for k-forms on a compact Riemannian manifolds, including a very cool decomposition theorem that generalizes the Helmholtz decomposition from vector calculus - Chapter 6 in Warner or Chapter 8 in G&Q2
- Geometric Theory of Automatic Differentiation : Short monograph that attempts to give a geometric interpretation to higher order automatic differentiation.
- Kahler Geometry : Neat concept in complex geometry with deep ties to Hodge Theory