18.965 | Fall 2004 | Graduate
Geometry of Manifolds

Calendar

LEC # TOPICS 
1 Manifolds: Definitions and Examples
2 Smooth Maps and the Notion of Equivalence

Standard Pathologies

3 The Derivative of a Map between Vector Spaces
4 Inverse and Implicit Function Theorems
5 More Examples
6 Vector Bundles and the Differential: New Vector Bundles from Old
7 Vector Bundles and the Differential: The Tangent Bundle
8 Connections

Partitions of Unity

The Grassmanian is Universal

9 The Embedding Manifolds in RN
10-11 Sard’s Theorem
12 Stratified Spaces
13 Fiber Bundles
14 Whitney’s Embedding Theorem, Medium Version
15 A Brief Introduction to Linear Analysis: Basic Definitions

A Brief Introduction to Linear Analysis: Compact Operators

16-17 A Brief Introduction to Linear Analysis: Fredholm Operators
18-19 Smale’s Sard Theorem
20 Parametric Transversality
21-22 The Strong Whitney Embedding Theorem
23-28 Morse Theory
29 Canonical Forms: The Lie Derivative
30 Canonical Forms: The Frobenious Integrability Theorem

Canonical Forms: Foliations

Characterizing a Codimension One Foliation in Terms of its Normal Vector

The Holonomy of Closed Loop in a Leaf

Reeb’s Stability Theorem

31 Differential Forms and de Rham’s Theorem: The Exterior Algebra
32 Differential Forms and de Rham’s Theorem: The Poincaré Lemma and Homotopy Invariance of the de Rham Cohomology

Cech Cohomology

33 Refinement

The Acyclicity of the Sheaf of p-forms

34 The Poincaré Lemma Implies the Equality of Cech Cohomology and de Rham Cohomology
35 The Immersion Theorem of Smale
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Fall 2004
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