| 1 |
Manifolds: Definitions and Examples |
(PDF) |
| 2 |
Smooth Maps and the Notion of EquivalenceStandard 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 |
ConnectionsPartitions 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 DefinitionsA 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 TheoremCanonical 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 CohomologyCech 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) |