MAT367 Differential Geometry

Assignments will be sent online, via Quercus, every two weeks. You will be asked to submit the solutions electronically, via Quercus.

Lecture notes:

• Introduction to Differential Geometry, book in progress by E. Meinrenken and G. Gross; its newer version with exercises solved is on Quercus.
• Lecture (March 19) on Cartan Calculus: Exterior differential, cohomology groups, Lie derivatives, pull-backs, and contractions of differential forms Exercises for Lecture March 19
  
• Lecture (March 20) on integration of differential forms and the Stokes theorem
• Lecture (March 27) on the winding number, degree of a map, linking number, and volume forms
• Bonus* (not required): Lecture (April 2-3) on homology groups, Euler characteristics, Hopf invariant, index of a zero of a vector field, the Poincare-Hopf theorem
• Bonus* (not required): Fun problems in Geometry

In addition, the following texts are recommended:

• Calculus on manifolds, notes by S. Yakoveno, 87pp.
• An introduction to differentiable manifolds and Riemannian geometry, by W.M.Boothby, Academic Press.
• Morse theory, by J.Milnor, ISBN 0691080089,

Additional sources:

• The Hopf Fibration: interactive visualization
• A note on the Fundamental Theorem of Algebra (lecture in detail)
• Huygens and Barrow, Newton and Hooke, by V.Arnold

• 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