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The geometry of Manifold increases your understanding of the differentiable manifolds and vector fields and forms. From this [course_title] you will be able to have a clear conception on Banach Manifolds, Sard’s Theorem, Whitney’s Embedding Theorems, Freudenthal Suspension Theorem, the de Rham theorem as well as Riemannian Manifolds to help you solve various problems efficiently.


This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer needs to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.


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Course Credit: MIT

Course Curriculum

Module: 01
Manifolds: Definitions and Examples 00:30:00
Smooth Maps and the Notion of Equivalence 00:45:00
The Derivative of a Map between Vector Spaces 00:30:00
Inverse and Implicit Function Theorems 00:45:00
More Examples 00:30:00
Module: 02
Vector Bundles and the Differential: New Vector Bundles from Old 00:30:00
Vector Bundles and the Differential: The Tangent Bundle 00:30:00
Connections 01:00:00
The Embedding Manifolds in RN 00:30:00
Sard’s Theorem 01:00:00
Module: 03
Stratified Spaces 00:45:00
Fiber Bundles 00:30:00
Whitney’s Embedding Theorem, Medium Version 00:30:00
A Brief Introduction to Linear Analysis: Basic Definitions 01:00:00
A Brief Introduction to Linear Analysis: Fredholm Operators 00:45:00
Smale’s Sard Theorem 00:45:00
Module: 04
Parametric Transversality 00:30:00
The Strong Whitney Embedding Theorem 00:45:00
Morse Theory 01:00:00
Canonical Forms: The Lie Derivative 00:30:00
Canonical Forms: The Frobenious Integrability Theorem 01:00:00
Module: 05
Differential Forms and de Rham’s Theorem: The Exterior Algebra 00:45:00
Differential Forms and de Rham’s Theorem: The Poincaré Lemma and Homotopy Invariance of the de Rham Cohomology 01:00:00
Refinement The Acyclicity of the Sheaf of p-forms 00:30:00
The Poincaré Lemma Implies the Equality of Cech Cohomology and de Rham Cohomology 00:45:00
The Immersion Theorem of Smale 00:30:00
Submit Your Assignment 00:00:00
Certification 00:00:00

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