Math 6520, Differentiable Manifolds
Fall 2017
Tuesday, Thursday in Malott 205
11:40 AM to 12:55 PM

Instructor: Bob Connelly (connelly@math.cornell.edu)
Office: 565 Malott
Office hours: Monday, Wednesday, Friday 1:30-2:15 in Malott 565
 
Last updated:  November 27, 2017

Prerequisites: A good course in linear algebra, point set topology, and analysis.



Lee will be the basic text of this course but there are other texts that have been used in the past.

    John Lee, Introduction to Smooth Manifolds
    William Boothby, Introduction to Differentiable Manifolds and Riemannian Geometry (in previous years)
    Lawrence Conlon, Differentiable Manifolds (in previous years)

The following texts also cover a lot of the same material, and especially the first one can serve as a more concret introduction, and it has a more accessible collection of examples and problems.  These should be available through the Math Library.
    Victor Guillemin and Alan Pollack, Differential Topology
    Abraham, Marsden, and Ratiu, Manifolds, Tensor Analysis and Appications

Minimum Syllabus

   1.     Manifolds, submanifolds. Immersions, embeddings and submersions.
   2.     Tangent bundles and tangent maps. Vector fields, derivations and the Lie bracket.
   3,     Sard’s theorem, easy Whitney embedding theorem.
   4.     Trajectories and flows of vector fields. Frobenius integrability theorem.
   5.     Connections, curvature and geodesics. Riemannian metrics, Levi-Civita connections.
   6.     Tensors, differential forms. Exterior derivative and Stokes’ theorem.

For  Aug. 22 to Sept 1, read Chapter 1 of Lee and hand in Problems:  1-6, 1-7, 1-8, 1-9.
For  Aug. 28 to Sept 7, read Chapter 3 of Lee and hand in Problems: 3-4, 3-6, and show that the tangent bundle of the 3-sphere is trivial.  (Hint:  Regard the 3-sphere as unit quaternions.)
For Sept. 14, read Chapter 4: Immersions, etc.  Problems: 4-6, 4-12, 4-13, Write your initials as an immersion of a circle in the plane, and compute its tangent winding number.
For Sept. 26, read Chapter 2.1 and 2.2 of Guillemin and Pollack, and do Exercises 1, 10, 11, in Secton 1, and Exercises 2, 6, 7 in Section 2.
For Oct. 3, read Chapter 6 of Lee, and do the following problems.
For Oct. 26, read Chapter 7 of Lee, and do problems, 7-4, 7-11, 7-16,  7-23,
Here is an outline of Sabastian's talk about the Jorndan separation theorem.
For Nov. 9 read Chapter 11, and hand in problems 11-1, 11-9, 11-16.

For the last assignment of the semester, do problem 16-6 in Lee on the Hairy Ball Theorem.  Due December 7, 2017.

For the week of Oct. 24 to Oct. 31, we had a take-home exam to be handed out in class and due on Oct. 31 in class.  Any source for your answers must be clearly and explicitly labeled.  Here is a corrected take-home exam.

Link to the Math Department.