Calculus on Manifolds

Outline

This course is about the differential geometry and topology of manifolds, which are higher dimensional generalization of curves (1-manifolds) and surfaces (2-manifolds), defined by way of parametrizations coming from some larger dimensional Euclidean space. Our focus will be on differential forms and integration on manifolds, including Stokes' theorem, which generalize and encompass multivariable calculus and analysis. Differential forms are smoothly varying alternating multilinear functionals on the tangent spaces of the manifold and remarkably encode global properties of the manifold via DeRham theory. We will also consider special kinds of smooth mappings between manifolds, namely immersions and submersions, and the intersections of submanifolds via transversality.

Our use of parametrizations and embeddings in higher dimensional Euclidean space to define manifolds allows us to not require point set topology as a pre-requisite. Ultimately, it turns out that this definition in terms of parametrizations is equivalent to the more abstract definition in terms of charts and atlases. We will follow the modern point of view on differential geometry and topology by emphasizing global aspects of the subject whenever possible.

This course is intended to be a precursor to graduate courses in differential geometry and topology. Thus more emphasis is placed on self-learning and lemmas and theorems will not typically be worked out in full detail in lectures. 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 some proofs and technical lemmas for you to read in the book (or not). I will state key lemmas and theorems and summarize the main points. You should expect to spend 1-3 hours a week reading the textbook each week. Some review of multivariable calculus and analysis will be required, which I will recap briefly in class.

The basic plan is to cover most of the material in chapters 1, 2, 4, the appendices of Guillemin-Pollack, and a bit more about DeRham cohomology. Time permitting, we will cover intersection theory and the Poincare-Hopf theorem.

• Smooth manifolds, smooth maps, diffeomorphisms
• Tangent vectors, tangent space, differentials
• Transversality and Sard's theorem
• Multilinear and Exterior Algebra
• Differential forms and Stokes' theorem
• Introduction to DeRham cohomology
• Classification of 1 manifolds
• Intersection theory (time permitting)
• The Poincare-Hopf theorem (time permitting)

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.