Math 225A:  Differentiable Manifolds

Fall 2020


This is the first quarter of a year-long sequence in geometry and topology.

Instructor: Ko Honda
Office: MS 7919
Office Hours: W 3-4pm or by appointment
E-mail: honda at math dot ucla dot edu


TA: Jason Schuchardt; office hours TBA; jason.sch at math dot ucla dot

Class Meetings:  I plan to record the lectures.

  • Lectures: MWF 2pm - 2:50pm on Zoom
  • Discussion: Th 2pm - 2:50pm on Zoom


  1. Review of advanced calculus (calculus on R^n); inverse and implicit function theorems.
  2. Differentiable manifolds and their maps.
  3. Tangent and cotangent bundles, vector bundles.
  4. Differential forms: tensor and exterior algebra, exterior differentiation, and Lie derivatives.
  5. Integration: Stokes' theorem, de Rham cohomology, and computations using Meyer-Vietoris sequences.
  6. Vector fields, distributions, Frobenius' theorem.


  • Knowledge of calculus on R^n, as presented in the first three chapters of Spivak's Calculus on Manifolds book.
  • This course requires more mathematical maturity than the average first-year graduate course in the mathematics department. 

There will be weekly problem sets; see the class schedule.  Homework is due on Fridays, although there may be some exceptional weeks.  The problem sets count for a large percentage of your total grade (approximately 70%).  You may work with others or consult other textbooks, but the homework you turn in must be written by you, in your own words, and you must cite your sources used and your collaborators!

Final examination

There will be a take-home final.  This will be approximately 30% of your final grade.

I will follow my Differential Geometry Course Notes.  The main reference is Lee's book, where you can find more details and examples.
  1. Differential Geometry Course Notes
  2. Lee, Introduction to Smooth Manifolds
  3. Spivak, A Comprehensive Introduction to Differential Geometry
  4. Tu, An Introduction to Manifolds
  5. Warner, Foundations of Differentiable Manifolds and Lie Groups
  6. Peter Petersen's notes
WARNING:  The course syllabus provides a general plan for the course; deviations may become necessary. 

Last modified: October 1, 2020.