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Geometry of Manifolds >> Content Detail



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

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

Partitions of Unity

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

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

Cech Cohomology
33Refinement

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

 








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