MIT OpenCourseWare

Browse materialsMIT 18.965 · Fall 2004

Geometry of Manifolds

Subjects: Algebra · Geometry · Linear Algebra · Mathematics

Art: Anonymous painter,15th century, Westerner and Arab practicing geometry 15th century manuscript, 15th… · Public domain

Instructor: Prof. Tomasz Mrowka
Term: Fall 2004
Level: Graduate
What's inside: 1 lecture note · 32 other files · 3 web pages
Materials open: MIT OCW

Start here

Start course: Lecture 1: Manifolds: Definitions and Examples Find a topic

Progress is saved in this browser.

Export or import progress

About this course

Source: MIT OpenCourseWare 18.965 Geometry of Manifolds, Fall 2004, Prof. Tomasz Mrowka — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

Materials are grouped the way MIT organized them. A verified, course-wide order has not been established here.

Before you start

Prerequisites are not listed here. Check MIT's own syllabus for this course.

How this course was organized

What's here, and what isn't

A lecture sequence of 21 items is confirmed. It stays as organized materials rather than a full course order.

Materials by section

Grouped by section, as MIT organized them. Not sequenced or ordered.

Lecture Notes

26 files

Assignments

MIT OpenCourseWare, Prof. Tomasz Mrowka, 18.965 Geometry of Manifolds, Fall 2004.

License: CC BY-NC-SA 4.0 · Edition: Fall 2004 · Open the original course on OCW ↗

Organized by RedLectern, independent of MIT and not endorsed by MIT.

Per-resource rights are recorded on each resource; All-Rights-Reserved items are link-only, never mirrored.

Geometry of Manifolds

Before you start

What's here, and what isn't

Materials by section

Lecture Notes

Assignments

Lecture 1: Manifolds: Definitions and Examples

Lecture 2: Smooth Maps and the Notion of Equivalence Standard Pathologies

Lecture 3: The Derivative of a Map between Vector Spaces

Lecture 4: Inverse and Implicit Function Theorems

Lecture 5: More Examples

Lecture 6: Vector Bundles and the Differential: New Vector Bundles from Old

Lecture 7: Vector Bundles and the Differential: The Tangent Bundle

Lecture 8: Connections Partitions of Unity The Grassmanian is Universal

Lecture 9: The Embedding Manifolds in RN

Lecture 12: Stratified Spaces

Lecture 13: Fiber Bundles

Lecture 14: Whitney’s Embedding Theorem, Medium Version

Lecture 15: A Brief Introduction to Linear Analysis: Basic Definitions A Brief Introduction to Linear Analysis: Compact Operators

Lecture 20: Parametric Transversality

Lecture 29: Canonical Forms: The Lie Derivative

Lecture 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

Lecture 31: Differential Forms and de Rham’s Theorem: The Exterior Algebra

Lecture 32: Differential Forms and de Rham’s Theorem: The Poincaré Lemma and Homotopy Invariance of the de Rham Cohomology Cech Cohomology

Lecture 33: Refinement The Acyclicity of the Sheaf of p-forms

Lecture 34: The Poincaré Lemma Implies the Equality of Cech Cohomology and de Rham Cohomology

Lecture 35: The Immersion Theorem of Smale