# MAT367 Differential Geometry

**Course information**

Code: MAT367S

Instructor: Marco Gualtieri

Class schedule: MWF 1-2 in SS 1071

TA office hours: W5-6 and R10-11 in BA6135

Instructor office hours: F2:30-3:30 in BA6260

Schedule changes: TBA

Teaching assistant: Mykola Matviichuk and

Term Exams: Feb. 26 and Mar. 26

Final Exam: TBA

Marking Scheme: 20% Homework (best 5/6), 20% Test1, 20% Test2, 40% Final.

**Assignments**

Assignments will be sent online, via Crowdmark. You will be asked to submit the solutions electronically, via Crowdmark. See the Syllabus.

Assignment 1 is now available on Crowdmark. It is due January 19th at 23:59.

**Main notes and suggested references**

There is no textbook; course notes will be provided here. These notes are based on a book in progress by Prof E. Meinrenken and G. Gross.

- Introduction
- Definition of a manifold
- Examples of manifolds
- Orientability, Topology, and construction methods
- Appendix on equivalence relations
- Smooth maps and examples
- The Hopf Fibration: interactive visualization
- Submanifolds
- Local diffeomorphisms, Submersions, and Immersions
- The tangent space and the tangent bundle
- Vector fields, the Lie bracket, and flows

However, I do recommend these well-known texts:

*An introduction to differentiable manifolds and Riemannian geometry*, by W. M. Boothby, Academic Press.*Introduction to Smooth Manifolds*, by J. M. Lee.- Milnor’s
*Morse theory*, ISBN 0691080089,

Title photo taken from Sketches of Topology

**Overview of topics we hope to cover**

- Manifolds, definitions and examples
- Smooth maps and their properties
- Submanifolds
- Vector fields and their flows
- Lie brackets
- Frobenius’ theorem
- Differential forms
- The exterior derivative
- Cartan calculus
- Integration and Stokes’ theorem for general manifolds
- Linking and winding numbers