MATH 6520 Differentiable Manifolds I

Prerequisites

Undergraduate analysis, linear algebra, and point-set topology.

Textbooks

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

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.

Optional Topics

The above syllabus takes up much of the available time.  Nevertheless, there are some notable omissions.  We suggest spending the remaining time giving an informal introduction to one or more of the following:

  1. Lie groups, Lie algebras, homogeneous spaces. (This may be kept very brief. Students who want to know more can take the Lie groups course, MATH 6500.)
  2. Classification of 1- and 2-manifolds.
  3. De Rham theory. (Requires some elementary homological algebra — snake Lemma, five lemma — which should be stated without proof. Many students have seen this in algebra or algebraic topology.)
  4. Transversality.
  5. Morse Theory.

Notes to Instructor

The course does not require much material from the other core courses. It can be taken by incoming graduate students with a solid background and by graduate students in applied mathematics, provided they are willing to accept the following results on faith:

  1. Implicit function theorem. (Almost all students know this.)
  2. Existence and uniqueness theorem for ODE. (Many students have seen a version of this. Some of the finer points need to be explained, such as smooth dependence on initial data and global existence for linear systems.)
  3. Multilinear algebra over fields. (Not all students know this. It may be necessary to explain the construction of tensor products and alternating algebras and state the basic properties.)
  4. A few basic results of measure theory and point-set topology.

We recommend to the instructor giving careful statements of these results when the need arises without going into the proofs.