Dror Bar-Natan: Classes: 2000-01:

# Fundamental Concepts in Differential Geometry

Dror Bar-Natan, drorbn@math.huji.ac.il, 02-658-4187.

Agenda: Thoroughly understand smooth manifolds, differential forms, Stokes' theorem, the de-Rham complex, and general position/transversality arguments.

Main textbook: Glen Bredon's Topology and Geometry.

Other books: Bott and Tu's Differential Forms in Algebraic Topology, Guillemin and Pollack's Differential Topology, Spivak's Calculus on Manifolds.

Classes: Sundays 12:00-14:00 and Tuesdays 16:00-17:00 at Mathematics 110.

Office hours: Sundays 14:00-15:00 in my office, Mathematics 309.

Problem Session: Tuesdays 17:00-18:00 at Mathematics 110, with Assaf Libman, asil@math.huji.ac.il.

The grade: The final grade will be a weighted average of the final exam grade f and the homework grade h, with weights 0.85f+0.15h if h>f and 0.93f+0.07h if h<f (i.e., the carrot is bigger than the stick). The responsibility for the homework grade is entirely in the hands of Assaf Libman; I am told that he will base his grade on the best 8 assignments that each of you will submit.

### By the week:

 Oct 29, 31 Dream: Course overview, review of multi-variable calculus: the differential, the chain rule, the inverse and implicit function theorems. Reality: We did all that, though certain parts may have been confusing. Homework: Exercise #1 - PS, PDF. Nov 5, 7 Dream: Differentiable manifolds: definition and elementary properties, examples. Reality: We are tiny miny behind, after we spent too much time over the formalities of certain examples. We still need to see why the "quilt" definition is equivalent to the "functional structures" definition. Homework: Exercise #2 - PS, PDF. Hints - PS, PDF. Nov 12, 14 Dream: Tangent vectors, differentials, the chain rule. Reality: Done. Homework: Exercise #3 - PS, PDF. Hints - PS, PDF. Nov 19, 21 Dream: Sard's theorem and consequences. Reality: I decided to skip Sard's theorem for now. Instead we have talked about the local structure of immersions and submersions, and unfortunately, due to the lecturer's confusion, some stuff will have to be repeated next week. Homework: Exercise #4 - PS, PDF. Nov 26, 28 Dream: Vector fields and flows, the tangent space, embedding in Euclidean space. Reality: We skipped vector fields and started talking about embeddings in Euclidean space. Homework: Exercise #5 - PS, PDF. Dec 3 I'll be away on December 5th. Dream: Tubular neighborhoods. Reality: We finished the discussion about embeddings in Euclidean space modulo a few "debts". We started returning the debts, leaving the most important ones, partitions of unity and Sard's theorem, as a homework assignment (the former) or for the future (the latter). Homework: Exercise #6 - Page 1, Page 2. Dec 12 I'll be away on December 10th. Dream: Approximation by smooth functions and applications. Reality: On December 10th Assaf replaced me and did Sard's theorem. Homework: Exercise #7 - PS, PDF. Dec 17, 19 Dream: Little on Lie groups. Reality: Homework: Exercise #8 - PS, PDF. Dec 26 December 24th is Hannukah. Dream: Fiber bundles and transversality. Reality: Homework: Exercise #9 - PS, PDF. Dec 31, Jan 2 Dream: Time for reality to catch up with the dreams. Reality: Homework: Exercise #10 - PS, PDF. Jan 7, 9 Dream: I'll be away. Reality: I was away! See what I did! Homework: Exercise #11 - PS, PDF. Jan 14, 16 I may be away on January 14th. Dream: Exterior algebra. Reality: Homework: Exercise #12 - PS, PDF. Jan 21, 23 Dream: Differential forms and Stokes' theorem. Reality: Homework: Exercise #13 - PS, PDF. Jan 28, 30 Dream: The de-Rham complex and the Poincare lemma. Reality: Homework: Exercise #14 - PS, PDF. Extra 1 Dream: Applications: div grad curl and all that, Maxwell's equations. Reality: Extra 2 Dream: More applications: the planimeter, degrees and the linking number, Moser's theorem. Reality: Feb 22 Final exam at 10:00AM, Mathematics 110, 3 hours. Definition, Moed A. June 4 Moed B. January 29, 2002 Moed C.