Geometry of Manifolds

Lecture Notes

LEC #	TOPICS	LECTURE NOTES
1	Manifolds: Definitions and Examples	(PDF)
2	Smooth Maps and the  Notion of Equivalence Standard Pathologies	(PDF)
3	The Derivative of a Map between Vector Spaces	(PDF)
4	Inverse and Implicit Function Theorems	(PDF)
5	More Examples	(PDF)
6	Vector Bundles and the Differential: New Vector Bundles from Old	(PDF)
7	Vector Bundles and the Differential: The Tangent Bundle	(PDF)
8	Connections Partitions of Unity The Grassmanian is Universal	(PDF)
9	The Embedding Manifolds in RN	(PDF)
10-11	Sard’s Theorem	(PDF)
12	Stratified Spaces	(PDF)
13	Fiber Bundles	(PDF)
14	Whitney’s Embedding Theorem, Medium Version	(PDF)
15	A Brief Introduction to Linear Analysis: Basic Definitions A Brief Introduction to Linear Analysis: Compact Operators	(PDF)
16-17	A Brief Introduction to Linear Analysis: Fredholm Operators	(PDF)
18-19	Smale’s Sard Theorem	(PDF)
20	Parametric Transversality	(PDF)
21-22	The Strong Whitney Embedding Theorem	(PDF)
23-28	Morse Theory	(PDF)
29	Canonical Forms: The Lie Derivative	(PDF)
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	(PDF)
31	Differential Forms and de Rham’s Theorem: The Exterior Algebra	(PDF)
32	Differential Forms and de Rham’s Theorem: The Poincaré Lemma and Homotopy Invariance of the de Rham Cohomology Cech Cohomology	(PDF)
33	Refinement The Acyclicity of the Sheaf of p-forms	(PDF)
34	The Poincaré Lemma Implies the Equality of Cech Cohomology and de Rham Cohomology	(PDF)
35	The Immersion Theorem of Smale	(PDF)