Differentiable Manifolds
Prof. Jo Nelson
Math 451/551
Spring 2025
Email:
jo [dot] nelson [at] rice [dot] edu
Discussion/Lectures: TTh 10:50 - 12.05 am
Location: TBA
Office Hours: TBA
References (free download from Rice ezproxy):
J. Lee, Intro to Smooth Manifolds, 2nd. Ed., Springer GTM.
Outline
The basic plan is to cover most of the material in chapters 1-19 of Lee's book (adding a few interesting things which are not in the book). My goal is for you to understand the basic concepts listed below and to feel confident about encountering manifolds in the wild (e.g. outside of the classroom). This material is all essential background for graduate level geometry and topology research. In class I will try to introduce the main ideas, explain where they come from, and demonstrate how to use them, with an emphasis on examples. I will tend to leave most proofs and technical lemmas for you to read in Lee's book (or not).
Assessment, % of Course Grade
Your grade will be based on homework (50%), one midterm (20%), and the final exam (30%) and attendence. There will be approximately 11 weekly homework assignments; you may drop your two lowest (or nonexistent) homeworks. To receive a passing grade you must complete and pass the assignments, attend at least 70% of the classes, and foster an atmosphere of collegiality. For an A you must also attend at least 90% of the classes (or have an excused absence), for a B you must attend at least 80% of the classes, for a C, 70%, and for a D, 60%.
Teaching Assistant
Evan Huang is the teaching assistant for this course. Evan will hold
a weekly office hour. Evan will
grade homework and I will grade the midterm and exam. I will also review your homeworks and read your weekly homework reflections.
Homework
There will be 11 homework sets and homework will count for 50% of the final grade. You must upload your homework to gradescope by 11pm on Mondays. Collaboration is encouraged but the write up of the solutions should be in your own words. Late homework will not be accepted without prior authorization from me. Your lowest two homework scores will be dropped. In the event of illness or family emergency I must be notified ideally at least 24 hours in advance and documentation from your magister or doctor may be requested.
Help
If you find yourself confused, please seek help sooner rather than
later. I will be available to answer questions during my office hours
as will Evan.
Schedule & Assignments
Date
Material Covered
Homework (Tuesdays)
1/9
Definition of topological and smooth manifold, examples.
1/11
Diffeomorphisms. Tangent vectors.
1/16
Tangent space.
Derivative of a smooth map between smooth manifolds.
Homework 1 LaTeX
Due 1/16
1/18
Local coordinates.
1/23
Vector fields and the tangent bundle.
Homework 2 LaTeX
Due 1/23
2/25
Immersions, embeddings, and submersions.
Short movie ``Outside In".
1/30
Embeddings and submanifolds I
Homework 3 LaTeX
Due 1/30
2/1
Embeddings and submanifolds II
2/6
Transversality
Homework 4 LaTeX
Due 2/6
2/8
Go to your Monday classes!!
2/13
"Generic" transversality results
2/15
The Lie bracket of two vector fields. The flow of a vector field.
Hopf Fibration and Video Takehome Midterm LaTeX
Due 2/20
2/20
Lie Algebras and Lie Groups
Midterm Due 2/20
2/22
Multilinear algebra and tensors
2/27
The cotangent bundle
Differentials revisited Homework 5 LaTeX
Due 2/27
2/29
Symmetric Tensors. Riemannian Metrics.
3/5
TM = T*M via musical isomorphisms
Homework 6 LaTeX
Due 3/5 3/7
Alternating tensors in gory detail
3/19
Differential forms in general.
Wedge product, pullback, and exterior derivative. Homework 7 LaTeX
Due 3/19
3/21
Lie derivatives
Commutator of two VFs = 0 iff their flows commute.
3/26
Orientations revisited.
Volume form on a Riemannian manifold Homework 8 LaTeX
Due 3/26
3/28
Integration of differential forms.
A one-form is exact iff its integral over every loop is 0.
4/2
Stokes' theorem.
Homework 9 LaTeX
Due 4/2
4/4
Overview of de Rham cohomology.
4/9
Mayer-Vietoris computations
Homework 10 LaTeX
Due 4/9
4/11
Homotopy Invariance
4/16
Singular cohomology
Homework 11 LaTeX
Due 4/16
4/18
Isomorphism between singular and de Rham cohomology
Finals Week
Takehome Final LaTeX
Final due 4/30
Additional Course Policies
Comportment Expectations. The Department of Mathematics supports an inclusive learning environment where diversity and individual differences are understood, respected, and recognized as a source of strength. Racism, discrimination, harassment, and bullying will not be tolerated. We expect all participants in mathematics courses (students and faculty alike) to treat each other with courtesy and respect, and to adhere to the Mathematics Department Standards of Collegiality, Respect, and Sensitivity as well as the Rice Student Code of Conduct. If you think you have experienced or witnessed unprofessional or antagonistic behavior, then the matter should be brought to the attention of the instructor and/or department chair. The Ombudsperson is also available as an intermediate, informal option, and contacting them will not necessarily trigger a formal inquiry.
Title IX Responsible Employee Notification. Rice University cares about your wellbeing and safety. Rice encourages any student who has experienced an incident of harassment, pregnancy discrimination or gender discrimination or relationship, sexual, or other forms interpersonal violence to seek support through The SAFE Office. Students should be aware when seeking support on campus that most employees, including myself, as the instructor/TA, are required by Title IX to disclose all incidents of non-consensual interpersonal behaviors to Title IX professionals on campus who can act to support that student and meet their needs. For more information, please visit safe.rice.edu or email titleixsupport@rice.edu.
Disability-related Academic Accommodations. In order to receive disability-related academic accommodations, students must first be registered with the Disability Resource Center (DRC). Students who may need accommodations in this course should give me a written letter from the DRC within the first two weeks. More information on the DRC registration process is available online at https://drc.rice.edu/. Registered students must present an accommodation letter to the professor before exams or other accommodations can be provided. Students who have, or think they may have, a disability are invited to contact DRC for a confidential discussion.