Math 550: Differentiable Manifolds II

David Dumas

University of Illinois at Chicago
Spring 2019

Primary text	J.M. Lee, Introduction to Smooth Manifolds. (Springer GTM, 2012) Free ebook through UIC
Lectures	MWF at 9am in 317 Taft Hall
Instructor email	david@dumas.io
Instructor office hours	MWF 11am (May 3 office hours end at 11:25am)
CRN	37861

About the course

Building on the foundational material from Math 549, we will discuss several aspects of the geometry and topology of smooth manifolds. The course will be divided into three roughly equal units focusing on the following topics:

• Smooth actions and quotients
• Bundles, connections, and curvature
• Symplectic manifolds

Supplementary texts

Most of the material we will cover can be found in the main textbook (Lee); however, we will occasionally cover material drawn from the supplementary texts listed below. These books may also be useful for students seeking a second exposition of material that is included in the primary text.

• F. Warner, Foundations of Differentiable Manifolds and Lie Groups. (Springer GTM, 1983)
• M. Spivak, A Comprehensive Introduction to Differential Geometry, Volume 1. (Publish or Perish, 1999) (Full text on archive.org)
• R. W. Sharpe, Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. (Springer GTM, 2000)
• A. Cannas da Silva, Lectures on Symplectic Geometry. (Online lecture notes; also available in Springer LNM series)
• D. McDuff and D. Salamon, Introduction to Symplectic Topology. (Oxford, 1999)
• C. Taubes, Differential Geometry: Bundles, Connections, Metrics and Curvature. (Oxford, 2011)
• S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Volume 1 (Wiley, 1996)
  (This is a reprint of the 1963 edition, which can also be used.)

Documents

• Course Syllabus
• Addendum to lecture 15: Twisted products
• Transformation formula for principal connection under gauge transformation
• Project topic suggestions

Problem sets

Problem sets and their due dates are listed below. Be sure to read the syllabus which has more detailed policies and requirements about your solutions to the problems. Unless otherwise noted, problem sets are due in class at the beginning of the lecture.

• Assignment due on Friday January 18: Read the syllabus
• Problem set 1, due Friday February 1 (due date changed due to lecture cancelation)
• Problem set 2 due Monday February 11
• Problem set 3 due Monday February 25
• Problem set 4 due Monday March 11
• Problem set 5 due Wednesday April 3   Updated March 17
• Problem set 6 due Monday April 15
• Problem set 7 due Monday April 29

Lecture topics

• Lecture 1 (Mon Jan 14): Intro: Smooth distributions and the Frobenius theorem (Lee, Chapter 19)
• Lecture 2 (Wed Jan 16): The local Frobenius theorem
• Lecture 3 (Fri Jan 18): Proof of the local Frobenius theorem
• Lecture 4 (Wed Jan 23): The global Frobenius theorem
• Lecture 5 (Fri Jan 25): Proof of the global Frobenius theorem
• Lecture 6 (Mon Jan 28): The Lie correspondence
• Lecture 7 (Wed Jan 30): (class canceled due to weather)
• Lecture 8 (Fri Feb 1): Topological group actions, properness (Lee, Chapter 21)
• Lecture 9 (Mon Feb 4): Smooth actions, the Quotient Manifold Theorem
• Lecture 10 (Wed Feb 6): Slices and tubes
• Lecture 11 (Fri Feb 8): Completion of proof of the QMT
• Lecture 12 (Mon Feb 11): Homogeneous spaces, properly discontinuous group actions
• Lecture 13 (Wed Feb 13): Fiber bundles
• Lecture 14 (Fri Feb 15): G-structures and principal bundles
• Lecture 15 (Mon Feb 18): Principal bundles as right G-spaces, associated bundles, cocycles
• Lecture 16 (Wed Feb 20): The frame bundle and associated bundles
• Lecture 17 (Fri Feb 22): Reduction of structure group
• Lecture 18 (Mon Feb 25): Smooth path connections, groupoids
• Lecture 19 (Wed Feb 27): Principal connections as splittings of TP and as 𝔤-valued forms
• Lecture 20 (Fri Mar 1): Horizontal lifts
• Lecture 21 (Mon Mar 4): Local connection forms
• Lecture 22 (Wed Mar 6): More on Maurer-Cartan forms; existence of principal connections
• Lecture 23 (Fri Mar 8): Curvature of principal connections
• Lecture 24 (Mon Mar 11): Parallel transport in principal bundles
• Lecture 25 (Wed Mar 13): Connections on vector bundles
• Lecture 26 (Fri Mar 15): Parallel transport in vector bundles
• Lecture 27 (Mon Mar 18): Curvature in vector bundles
• Lecture 28 (Wed Mar 20): The exterior covariant derivative
• Lecture 29 (Fri Mar 22): Curvature is the derivative of parallel transport
• Lecture 30 (Mon Apr 1): Equivalent characterizations of flatness
• Lecture 31 (Wed Apr 3): Correspondence between flat vector bundles and twisted products
• Lecture 32 (Fri Apr 5): Equivalence of vector and principal connections
• Lecture 33 (Mon Apr 8): Equivalence of vector and principal connections (continued)
• Lecture 34 (Wed Apr 10): Symplectic vector spaces
• Lecture 35 (Fri Apr 12): Symplectic manifolds
• Lecture 36 (Mon Apr 15): Hamiltonian vector fields and the Poisson bracket
• Lecture 37 (Wed Apr 17): The Darboux Theorem
• Lecture 38 (Fri Apr 19): Examples of symplectic manifolds and Hamiltonian systems
• Lecture 39 (Mon Apr 22): Symplectic and Hamiltonian group actions
• Lecture 40 (Wed Apr 24): Examples of moment maps
• Lecture 41 (Fri Apr 26): Marsden-Weinstein-Meyer quotient theorem
• Lecture 42 (Mon Apr 29): Student presentations
• Lecture 43 (Wed May 1): Student presentations
• Lecture 44 (Fri May 3): Student presentations

Links and resources

Here we collect citations for further reading about course material (beyond the course texts) and links to relevant online materials. Links marked ^* are available with UIC's subscription by using on-campus computers or the library's proxy service.

• Groupoids
  • Briefly mentioned in S. MacLane, Categories for the Working Mathematician, which is a good general resource if categorical algebra arises in your work but is not your main focus area.
  • This nice survey discusses applications and has many references: R. Brown, From Groups to Groupoids: A Brief Survey, Bulletin of the London Mathematical Society 19 (1987) 113-134.
• Path groupoid, path connections, thin homotopy
  • My treatment in lecture most closely follows: E. E. Wood, Reconstruction theorem for groupoids and principal fiber bundles, International Journal of Theoretical Physics 36 (1997) 1253-1267.^*
  • Smooth thin homotopy was introduced and studied in relation to connections in: A. Caetano and R. F. Picken, An Axiomatic Definition of Holonomy, International Journal of Mathematics 5 (1994) 835-848.. (A preprint version is available in full text from CERN.)
• Equivalence of vector and principal connections
  • Sections 18 and 19 of: P. W. Michor, Topics in Differential Geometry, American Mathematical Society, 2008.

LaTeX

Students who are learning TeX / LaTeX and who want to typeset their homework may find this template helpful:

• LaTeX homework template

Up: Home page of David Dumas