Math 225A:  Differentiable Manifolds

Fall 2021


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

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


TA: Eilon Reisin-Tzur; office hours TBA; ereisint at math dot ucla dot

Class Meetings:

  • Lectures: MWF 2pm - 2:50pm in MS 5117
  • Discussion: Tu 2pm - 2:50pm in Franz Hall 2288

COVID-19 Policy:

Keeping our community safe depends on each of us following the latest UCLA health and safety guidelines. Things might change, but for the moment you:

  • Are responsible, regardless of vaccination status, for wearing an approved mask that fully covers our nose and mouth for the duration of class, office hours, or other course-related activity.  Disposable masks are available at the Wooden Center for anyone unable to obtain a mask or who has forgotten to bring one to campus.  Appropriate masks include two-ply woven fabric masks, surgical masks, non-woven KN95 masks, and N95 respirators. Please note that scarves, balaclavas/ski masks, single-layer fabric masks and neck gaiters, bandanas, and turtleneck collars are not adequate. For those that have a medical reason not to wear a mask, you can contact the Center for Accessible Education (CAE) to have this exception approved and sent to instructors.
  • Must be fully vaccinated or have submitted an exception request.  Unvaccinated students with pending or approved exceptions must comply with twice-weekly testing.
  • Are required to complete daily symptom checks prior to coming to campus, regardless of vaccination status, and must stay home if you are not cleared by the symptom survey and/or are advised by the Exposure Management Team to quarantine or isolate.
  • Will refrain from eating or drinking in the classroom. If you need to take a sip of water or eat something quickly for medical reasons, please step outside the room to do so.

Please stay home if sick or potentially exposed. Email me if you need to stay home, and we will arrange for you to access class recordings/notes.

The flip side of this requirement is that I also cannot come in even if I have mild cold symptoms such as a headache, runny nose, or sore throat.  I think it's likely this will happen at some point in the quarter because I catch colds pretty often.  In that case I'll email everyone in the class and we'll have to switch to Zoom for a lecture or two.

Be advised that refusal to comply with current campus directives related to COVID-19 mitigation will result in dismissal from the classroom and referral to the Office of Student Conduct. If you have any questions or concerns about UCLA’s COVID-19 protocol, go to


  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: September 15, 2021.