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Geometry of Manifolds >> Content Detail



Lecture Notes



Lecture Notes

LEC #TOPICS LECTURE NOTES
1Manifolds: Definitions and Examples(PDF)
2Smooth Maps and the  Notion of Equivalence

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

Partitions of Unity

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

A Brief Introduction to Linear Analysis: Compact Operators 
(PDF)
16-17A Brief Introduction to Linear Analysis: Fredholm Operators(PDF)
18-19Smale's Sard Theorem(PDF)
20Parametric Transversality(PDF)
21-22The Strong Whitney Embedding Theorem(PDF)
23-28Morse Theory(PDF)
29Canonical Forms: The Lie Derivative(PDF)
30Canonical 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
(PDF)
31Differential Forms and de Rham's Theorem: The Exterior Algebra(PDF)
32Differential Forms and de Rham's Theorem: The Poincaré Lemma and Homotopy Invariance of the de Rham Cohomology

Cech Cohomology
(PDF)
33Refinement The Acyclicity of the Sheaf of p-forms(PDF)
34The Poincaré Lemma Implies the Equality of Cech Cohomology and de Rham Cohomology(PDF)
35The Immersion Theorem of Smale(PDF)

 








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