Math 518, Differentiable Manifolds I

Fall 2014

Course Description

This is a first graduate course on smooth manifolds, introducing various aspects of their topology, geometry, and analysis. We will start at the beginning with the definition of a smooth manifold, look at some examples, and then explore the basic associated objects, including submanifolds, tangent vectors, bundles, and derivatives. We will apply the inverse function theorem to geometric issues like transversality, and then look at vector fields, associated flows, and the Lie derivative. Differential forms on manifolds will also be a focus, including how to differentiate and integrate them. Time permitting, we might look at the very basics of Lie groups, foliations (the Frobenius theorem), Morse theory, or de Rham cohomology. In addition to treating the foundations of the subject carefully, this course aims to emphasize examples and geometric intuition throughout.

Prerequisites: A good understanding of basic real analysis in several variables (e.g. the inverse and implicit function theorems) and some knowledge of metric spaces or point-set topology.


Here are some recommended references; the first one is the official text, but the others are good too. For the first three, the book title is linked to a PDF version which is free to all U of I folks.


Your course grade will be based on:

Homework Assignments

Lecture notes

Here scans of my lecture notes, in PDF format.

  1. August 25. Introduction. Actually did pages 1-4.
  2. August 27. Smooth manifolds. Actually did pages 1-3, page 4 up to the lemma, and the top part of page 7.
  3. August 29. Smooth maps and diffeomorphisms. Did pages 1-5 with some brief discussion of 6.
  4. Sept 3. Tangent spaces. Did pages 1-4 with a somewhat extended discussion of how this will be helpful.
  5. Sept 5. Derivations as tangent vectors. Did through first lemma on page 6.
  6. Sept 8. More on tangent spaces. Did everything except the discussion of germs.
  7. Sept 10. Immersions, embeddings, and covering maps. Everything with the exception of the examples at the very bottom of page 4.
  8. Sept 12. Covering maps and submersions. Did everything except the statement of the inverse function theorem.
  9. Sept 15. Inverse Function Theorem. Did everything except the proof that the inverse is smooth.
  10. Sept 17. Immersions and submersions in local coordinates. Did pages 1-3.
  11. Sept 19. Preimages of submersions. Did 1-3 and the high points of 4-5.
  12. Sept 22. Tangent bundles and vector fields. Did 1-5 through the statement of the theorem.
  13. Sept 24. Lie groups. Did 1-4.
  14. Sept 26. More on Lie groups. Did 1-4 and very top of 5.
  15. Sept 29. Lie groups: subgroups and actions. Did everything, though last page was rushed.
  16. Oct 1. Vector fields integral curves, and flows. Did everything, though last page was rushed.
  17. Oct 3. Lie algebras, Lie derivatives, and Lie brackets. Did everything, though last page was rushed.
  18. Oct 6. Equality of the Lie bracket and the Lie derivative. Did 1-4.
  19. Oct 8. Lie algebras of Lie groups. Did 1-4, but ended 5 minutes early.
  20. Oct 10. 1-parameter subgroups and the exponential map. Did everything.
  21. Oct 15. Prelude to integration. Did everything, almost ran out of material.
  22. Oct 17. Covector fields. Did everything.
  23. Oct 20. Integrating covector fields. Did 1-4.
  24. Oct 22. Riemannian metrics. Did 1-5.
  25. Oct 24. Riemannian geometry. Did 1-3 and a very condensed version of 4-5.
  26. Oct 27. Tensors and k-forms. Did 1-5.
  27. Oct 29. Differential forms. Through definition of orientation on page 5.
  28. Oct 31. Orientations of manifolds. Did everything.
  29. Nov 3. Partitions of Unity. Did everything, almost ran out of material.
  30. Nov 5. Integration on Manifolds. Covered everything; the missing page 4 does not exist.
  31. Nov 7. Exterior differentiation. Through very top of page 5.
  32. Nov 10. Lie derivatives of forms; statement of Stokes theorem. Did everything.
  33. Nov 12. Proof of Stokes theorem. Did 1-3 and 5.
  34. Nov 14. No class. Read the section The Divergence Theorem in Chapter 16 of Lee (very bottom of page 422 through the top of page 426).
  35. Nov 17. Calculus to cohomology. Did 1-(top of 4), plus subsequent items 2 and 3.
  36. Nov 19. Properties of de Rham cohomology. Did through top half of 4.
  37. Nov 21. Homotopies and cohomology. Did everything.
  38. Dec 1. Mayer-Vietoris by example. Did 1-4.
  39. Dec 3. The Mayer-Vietoris sequence. Did 1-(very top of 4) but purposely ended 10 minutes early.
  40. Dec 5. Degrees of maps of spheres. Did everything.
  41. Dec 8. Applications of cohomology. Did everything with time left for ICES forms.
  42. Dec 10. Poincaré duality. Did 1-4 and the 5 really fast.

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