Harvard College/GSAS: 0372
Location: Science Center 411
Meeting Time: M., W., F., at 10
Exam Group: 5
Prerequisite: Mathematics 132 or equivalent.
Office Hours: Th 12:30 PM - 2 PM and Fr 1:30 - 3:30 PM.

This class covers the basics of smooth manifolds and their differential geometry. Topics include connections and curvature (on principle and tangent bundles), Lie groups, de Rham cohomology, and other topics as time allows.

1. Wed, Sept 3. Introduction to the class. Manifolds and structures.
2. Fri, Sept 5. Another introduction to geometry, smooth maps. Corrections in red.
   An addendum showing equivalence of a topological manifold being: paracompact with countably many components, second countable, admitting compact exhaustions.
3. Mon, Sept 8. Tangent spaces as derivations (finally).
4. Wed, Sept 10. The tangent bundle.
5. Fri, Sept 12. The tangent bundle continued; vector bundles and sections.
6. Mon, Sept 15. Lie algebras.
7. Wed, Sept 17. Derivations are vector fields. Cotangent bundles.
8. Fri, Sept 19. Making vector bundles.
9. Mon, Sept 22. Categories and Mfld-enriched categories as the pattern for making new vector bundles out of old.
10. Wed, Sept 24. End of constructing new vector bundles. DeRham cohomology.
11. Fri, Sept 26. Outline of characteristic classes and applications. Connections.
12. Mon, Sept 29. Connections.
13. Wed, Oct 1. Connections exist. Convexity of space of connections. Toward curvature.
14. Fri, Oct 3. What does flatness mean?
15. Mon, Oct 6. Structure equation. Curvature in local coordinates.
16. Wed, Oct 8. Curvature some more. Also, an Addendum: Sections of Hom and tensor bundles.
17. Fri, Oct 10. Invariant polynomials.
18. Wed, Oct 15. More on invariant polynomials. Closedness of characteristic classes.
19. Fri, Oct 17. Homotopy invariance of deRham cohomology. Independence of char. classes on connection.
20. Mon, Oct 20. Notes one: Independence of characteristic class on connection. Notes two: Definition of Pontrjagin and Chern classes.
21. Wed, Oct 22. Part I: More complex vector bundles. Part II: Whitney sum/product formula.
22. Fri, Oct 24. Integration of differential forms and Stokes's Theorem.
23. Mon, Oct 27. Proof of Stokes's Theorem.
24. Wed, Oct 29. Robbie Guest lecture. Metrics. What Hiro thinks you'll need for the homework.
25. Fri, Oct 31. Robbie Guest lecture. Notes for Robbie's first lecture on Stokes's Theorem and Riemannian manifolds. Notes for Robbie's second lecture, fundamental theorem of Riemannian geometry.
26. Mon, Nov 3. Pfaffian and Euler class. Statement of Gauss-Bonnet-Chern Theorem
27. Wed, Nov 5. Proof of Gauss-Bonnet-Chern Theorem.
28. Fri, Nov 7. Compatible connections on arbitrary vector bundles. Proof of Fundamental Theorem of Riemannian Geometry.
29. Mon, Nov 10. Ehresmann connections; connections as horizontal, scale-invariant distributions.
30. Wed, Nov 12. Flows, bracket, Lie derivatives
31. Fri, Nov 14. Curvature, geodesics.
32. Mon, Nov 17. Exponential map.
33. Wed, Nov 19. Sectional curvature and local uniqueness of geodesics
34. Fri, Nov 21. Hadamaard's Theorem and Covering Spaces.
35. Mon, Nov 24. Hadamaard's Theorem and Jacobi Fields.
36. Mon, Dec 1. Finish proof of Hadamaard's Theorem.
37. Wed, Dec 3. Final lecture.

1. Homework one, due Mon, Sept 8. Do Problem 1 from Batch A. Corrections in blue.
2. Homework two, corrections in blue, due Mon, Sept 22.
3. Homework three, also due Mon, Sept 22.
4. Homework four, due Mon, Sept 29.
5. Homework five, due Mon, Oct 6.
6. Homework six, due Mon, Oct 20. (Note you have two weeks for this one.)
7. Homework seven, due Mon, Oct 27.
8. Homework eight, due Mon, Nov 3.
9. Homework nine, due Mon, Nov 10.
10. Homework ten, due Mon, Nov 17. Note that most problems are optional, and that the long-looking required problems are actually not so bad.
11. Homework eleven, due Mon, Nov 24.
12. No Homework Twelve. Take-home final will be given out Dec 3 or 4, due by the last day of finals period.
13. Final exam. Read the exam for due date.

There will be no textbook that we closely follow in this class. I will post online hand-written notes after every lecture. Regardless, below are a list of resources you should feel free to consult.

For an introduction to smooth manifolds with more of a view towards geometry (these are the books most appropriate for our class):

• Introduction to Smooth Manifolds by John Lee.
• Manifolds and Differential Geometry by Jeffrey Lee.
• Foundations of Differential Manifolds and Lie Groups by Frank W. Warner. (The source Hiro learned from as a student.)
• The appendix to Characteristic Classes by J. Milnor and J. Stasheff.

For an introduction with a focus on Riemannian geometry (i.e., the differential geometry of manifolds with positive-definite metrics, as opposed to say, Lorentzian):

• Differential and Riemannian Manifolds by Serge Lang.
• Riemannian Geometry by Manfredo P. do Carmo. (The source Hiro learned from as a student.)
• Morse Theory by Milnor.

For a basic introduction to smooth manifolds and their basic topology:

• Differential Topology by Guillemin and Pollack. (The source Hiro learned from as a student.)
• Topology from the Differentiable Viewpoint, by Milnor.
• Differential Forms in Algebraic Topology by Bott and Tu.

Other resources

• Differential geometry: Its past and its future. Chern's speech from the 1970 ICM.