# Math 518 - Differentiable Manifolds I - Fall 2020

## (section X1)

NOTE: This is the public version of the course webpage. More information is available on the moodle course webpage, accessible only to registered students.

The notion of differentiable manifold makes precise the concept of a space which locally looks like the usual euclidean space Rn. Hence, it generalizes the usual notions of curve (locally looks like R1) and surface (locally looks like R2). 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., differential and integral 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.
The main goals of this course are:
• Understand what are manifolds and maps between them, and learn how to construct them;
• Understand the symmetry groups of manifolds (Lie groups) and their infinitesimal versions (Lie algebras), and how to work with them;
• Study differential and integral calculus on manifolds, using objects called differential forms.
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: See the moodle course webpage for weekly zoom sessions or contact the lecturer via email for other arrangements
Class meets: This course will be held on-line via zoom. See the moodle course webpage for details and recordings of the lectures
Prerequisites: Restricted to Graduate Students; Undergraduate students may register with approval; The course assumes some background in basic abstract algebra and point set topology.

## Course contents:

• 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.

## Recommended Textbooks:

You can find the written version of the lecture notes here. The following two textbooks are highly recommended.
• John M. Lee, Introduction to Smooth Manifolds, Springer-Verlag, GTM vol 218, 2nd Ed, 2012. (there is an e-version of this book; see the contents and first chapter here).
• Michael Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1, (3rd edition) Publish or Perish, 2003.

There will be weekly homework (40%), 1 midterm (20%) and a final exam (40%). Details are given in the moodle course webpage.

## Homework Assignments

These will only be posted in the moodle course webpage.

## Emergency information for students in Mathematics courses

For important emergency information related to fires, tornados or active threats, please look at the following leaflet.

Last updated July 18, 2020