MAT367 Differential Geometry

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.

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
• Differential forms

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

• 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