Differentiable Manifolds

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).

• Smooth manifolds, smooth maps, diffeomorphisms (Lee 1-2)
• Tangent vectors, tangent space, differential, tangent bundle (Lee 3)
• Transversality and Sard's theorem (Guillemin and Pollack, Lee 6)
• Vector fields and Lie bracket (Lee 8)
• Lie groups and Lie algebras (Lee 9)
• The cotangent bundle (Lee 10-11)
• Tensors and Riemannian metrics (Lee 12-13)
• Differential forms and Stokes' theorem (Lee 14-16 )
• Flows and the Lie derivative
• DeRham cohomology (Lee 17-18)
• Distributions and foliations (Lee 19)

Additional Course Policies

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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.