Math 102: Foundations of Smooth Manifolds

Fall 2011


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