Math 518 - Differentiable Manifolds I - Fall 2020
(section XXX)
The notion of differentiable manifold makes precise the concept of a space which locally looks like the usual euclidean space R^{n}. Hence, it generalizes the usual notions of curve (locally looks like R^{1}) and surface (locally looks like R^{2}). This course consists of a precise study of this fundamental concept of Mathematics and some of the constructions associated with it: for example, much of the infinitesimal analysis (i.e., calculus) extends from euclidean space to smooth manifolds. On the other hand, the global analysis of smooth manifolds requires new techniques and even the most elementary questions quickly lead to open questions.
If you would like to have a preview of this course and experience a taste of it, you may wish to watch an old recording of 3 lectures by Fields medalist John Milnor.
Lecturer: Rui Loja Fernandes
Email: ruiloja (at) illinois.edu
Office: 346 Illini Hall
Office Hours: TBD
Class meets: TBD
Prerequisites: Restricted to Graduate Students; Undergraduate students may register with approval.
In this page:
- Class will meet for the first time on Monday, August 24.
- Foundations of Differentiable Manifolds. Differentiable manifolds and differentiable maps. Tangent space and differential. Immersions and submersions. Embeddings and Whitney's Theorem. Foliations. Quotients.
- Lie Theory. Vector fields and flows. Lie derivatives and Lie brackets. Distributions and Frobenius' Theorem. Lie groups and Lie algebras. The Exponential map. Transformation groups.
- Differential Forms. Differential forms and Tensor fields. Differential and Cartan Calculus. Integration on manifolds and Stokes Formula.
You can find my lecture notes here, but the following two textbooks are highly recommended.
Recommended Textbooks:
- John M. Lee, Introduction to Smooth Manifolds, Springer-Verlag, GTM vol 218, 2003. See the contents and first chapter here.
- Michael Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1, (3rd edition) Publish or Perish, 1999.
There will be weekly homework, 1 midterm and a final exam. All exams/midterms will be closed book.
- Homework and in class participation (40% of the grade): Homework problems are to be assigned once a week. They are due the following week, at the beginning of the Friday class. No late homework will be accepted. The two worst homework grades will be dropped. On Fridays, one homework problem will be discussed in class, with the participation of the students and this will be taken into account for the grade.
- Midterm (20% of the grade): The midterm will take place on Friday, October 11, in the regular classroom (the date is subject to change).
- Final Exam (40% of the grade): You have to pass the final to pass the course. According to the non-combined final examination schedule it will take place 7:00-10:00PM, Tuesday, December 17, in the regular classroom.
Sections of the Lecture Notes covered:
(PDF files can be viewed using Adobe Acrobat Reader which can be downloaded for free from Adobe Systems for all operating systems.)
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Last updated March 1, 2020