MAT 569: Tuesday and Thursday 11:30am-12:50pm, Harriman Hall 115

MAT 569: Differential Geometry II

Course Instructor: Mark McLean

Tuesday and Thursday 11:30am-12:50pm. From March 30th onwards, this course will be taught online through Zoom. The Zoom meeting code will be on blackboard. You can also email me for the code.

Homework

For this course, you are required to do a small amount of homeork. I have set some questions here.

Homework (Due May 9th, 2020).

Lecture Notes

Bochner Technique
Gromov Hausdorff Convergence
Hölder Spaces and Schauder Estimates
Norms and Convergence of Manifolds
Geometric Applications
Harmonic Norms and Ricci Curvature
Sectional Curvature Comparison II

Videos

Lecture March 31 2020
Lecture April 2 2020
Lecture April 7 2020
Lecture April 9 2020
Lecture April 14 2020 (Access Password: x2@%4#=g).
Lecture April 16 2020 (Access Password: j4*0d55%).
Lecture April 21 2020 (Access Password: 5A$98*k).
Lecture April 23 2020 (Access Password: Password: 6P*m?6$B).
Lecture April 28 2020 (Access Password: 5u=$6V7+).
Lecture April 30 2020 (Access Password: 1a=$072!).
Lecture May 5 2020 (Access Password: 7T%b$*Ib).
Lecture May 7 2020 (Access Password: 3Q#!1Qg%).

Syllabus

We will hopefully cover the following topics:

The Bochner Technique.
Symmetric Spaces and Holonomy Groups.
Ricci Curvature and distance and volume bounds and the Cheeger-Gromoll splitting theorem.
Convergence (Gromov-Hausdorff and smooth convergence).

There will be homework for this course. However there will be no final exam.

Textbook

Peter Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 2006, ISBN 978-0-387-29403-2.

Prerequisites

You should be familiar with the contents of the differential geometry I course MAT 568. This course covered the first six chapters of Petersons book. In particular, metrics, geodesics, covariant derivatives (connections) and curvature, leading to the Cartan-Hadamard theorem on the universal cover of manifolds with negative sectional curvature and Myers' theorem on the diameter of manifolds of positive sectional curvature.

Office Hours via Zoom: Fri 1pm-2pm, Tuesday 1pm-2pm and Wednesday 10:30am-11:30am.
The Zoom meeting code will be on blackboard. You can also email me for the meeting code.

Old Syllabus before March 30

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