Math 102: Foundations of Smooth Manifolds
Fall 2011
Date |
Topic |
Reading |
HW |
| Sept. 21 | What is a topological Maniolds? | Boothby Chp. I; Lee: Chp. 1 |
|
| Sept. 23 | What is a differentiable structure? | Boothby: III.1-2; Lee: Chp. 1 |
|
| Sept. 26 | The Grassmann manifold | Boothby: III.2; Lee: Chp. 1 |
|
| Sept. 28 | New Manifolds from old: taking quotients |
Boothby: III.3 | HW1 |
| Sept. 30 | Smooth Functions; Lie Groups | Boothby III.4 | |
| Oct. 3 | Covering Maps; Universal Cover of a Lie group | Lee p. 40- 45 | |
| Oct. 5 | Rank of Maps & Submanifolds | Boothby III.4-5 | HW2 |
| Oct. 7 | Tangent Space of Euclidean Space & derivations | Boothby IV.1-2; Lee Chp. 4 |
|
| Oct. 10 | Tangent Space to a manifold and the differential |
Boothby IV.1-3; Lee Chp. 4 | |
| Oct. 12 | the differential of a map & smooth vector fields | Boothby IV.1-3; Lee Chp. 4 | HW3 |
| Oct. 14 | smooth vector fields & local flows | Boothby IV.1-IV.3 | |
| Oct. 17 | Group Actions; One-parameter Group Actions | Boothby IV.3 | |
| Oct. 19 | Local one-parameter group actions | Boothby IV.3 | |
| Oct. 21 | Existence & Uniqueness Theorem; Local one-parameter subgroups associated to smooth vector fields | Boothby IV.4 | |
| Oct. 24 | Correspondence Between local one-parameter group actions & vector fields | Boothby IV.4 | |
| Oct. 26 | Escape Lemma; Regular & Singular Points; Canonical Form of Flow near regular points; Complete Vector Fields | Boothby IV.5- IV.6 | |
| Oct. 28 | Left-invariant Vector Fields are complete; One-parameter Subgroups & Left-invariant vector Fields; the Exponential Map | Boothby IV.6 | HW4 |
| Oct. 31 | The Exponential Map & the closed subgroup theorem; The Exponential Map on GL_n(R); Lie Bracket | Boothby IV.6 - IV.7 | |
| Nov. 1 | Lie Bracket & Lie Algebra | Boothby IV.7 | |
| Nov. 2 | Lie Bracket & the Lie Algebra | Boothby IV.8 | |
| Nov. 4 | Lie Derivative | Boothby IV.7 | |
| Nov. 7 | Lie Derivative & The canonical form for commuting vector fields | Boothby IV. 8 | |
| Nov. 8 | Frobenius' Theorem & Systems of linear PDEs | Lee Chapter 19 | |
| Nov. 9 | Frobenius' Theorem & Systems of PDEs | Lee Chapter 19 & Boothby IV.9 | |
| Nov. 11 | Vector Bundles & the definition of the Cotangent Bundle | Lee Chp. 5 | HW5 |
| Nov. 14 | Cotangent Bundle, differentials & line integrals | Lee Chp. 5, Boothby V.1 | |
| Nov. 15 | Tensor Algebra & Tensor Fields | Lee Chp. 11, BoothbyV.5-6 | |
| Nov. 16 | Tensor Fields & Riemannian Metrics | Lee Chp. 11, BoothbyV.2 | |
| Nov. 18 | Parittions of Unity & the Existence of Riemannian Metrics;Alternating Tensors and differential forms |
Lee Chp. 11 Boothby V.2-4; Lee Chp. 12
|
|
| Nov. 21 | Exterior Differentiation; Orientation | Lee Chp. 12 & 13 | |
| Nov. 22 | Orientation on Manifolds | Lee Chp. 13 | |
| Nov. 28 | Orientation on Manifolds; Integration | Lee Chp. 13 | HW6 |
| Nov. 29 | Integration & Stokes' Theorem | Lee Chp. 14 | |
| Nov. 30 | Integration & Stokes' Theorem | Lee Chp. 14 |
Last Updated 11 November, 2011