Here is the syllabus.
Here is a section on surface theory containing a list of basic formulas that will be explained and derived in the first week or two of the course (before beginning the book by Lee), and a set of exercises.
Here is the solution to problem 2-6b in Lee. Here is the solution to problem 4-7.
The theorem that Lie derivative equals Lie bracket is proved here. (The proof in the book uses a theorem that we did not cover.)
Here is a set of notes on connections on a manifold (in the sense of Koszul).
Chapter 1: Read pages 1-8 (middle) and 11(middle)-22(middle). Do problem 5 on p. 28. For part (d), it is enough to show that one hemisphere projection is compatible with one stereographic projection.
Chapter 2: Read pages 30-37 and pages 49-55 (through the existence theorem for bump functions). Do problems 1, 3, 4, and 6 on page 57.
Chapter 3: Read pages 60-73(middle) and 75-78. Do exercises 1 and 2 on pages 65 and 66, and problems 1, 4, and 8 on pages 78-79.
Chapter 4, 5, 17, and 18: Read pages 80-92 (omitting manifolds with boundary on page 89), pages 103-109(middle), pages 434-440(middle), and pages 464-470. All of this is concerned with vector fields on a manifold. Do problems 5, 7 (difficult--see hints), 11, 14 on pages 101-102 and problems 1 and 5 on page 491.
Chapter 6: Read all. Do problems 5, 9, and 10 on pages 151-153. (I said read all, but I have decided not to cover pages 143-150 except for the definition of exact.
Chapter 11: Read pages 260-263 and from the middle of page 267 to 285. Do problems 10, 13, and 14 on pages 287-289. For problem 13, the definition of an immersion is found on page 156 in the second paragraph (the pushforward map F-lower-star is injective, that is, one-to-one, at each point).
Chapter 12: Read pages 292-306. Do problems 2, 3, 5, 6, 7 on pages 319-320.
Chapter 13 and 14: Read pages 324-328 and 349-354 and 359. We will study only the definitions. We will not do the proofs.
Chapters 7 and 8: Read pages 155-159 (statement of inverse function theorem), page 163 (statement of rank theorem), and pages 166-168 (on constant-rank maps). Read pages 174-177, pages 180 (starting at Level Sets) to page 182 (ending with Cor. 8.9), and pages 186-187 (immersed submanifolds).