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