John Lee "Smooth Manifolds" (for example).
This is a graduate class on the foundations of differential geometry.
See the
syllabus for more information about the course.
Time: M/W/F 2:30 PM -- 3:20 PM.
Location: KCB 201
Course materials
In addition to the text of Lu, I will post some standalone notes here as they become relevant.
I will also try to post my handwritten lecture notes from each class (see course log).
Additionally, while we only have one official textbook, the following list of references below are excellent (albeit cover very different topics).
Sometimes, these topics may align with the topics of the course, so that is why I am listing them here.
Differential forms . These are notes from a class taught by Victor Guillemin at MIT. Notes written by Peter J. Haine.
Notes on bordism , by Dan Freed. Great source for background on bundles, principal bundles, and characteristic classes.
Course log
September 3: Inner products and Riemannian metrics; notes.
September 5: Isometries, tensor fields, covariant derivatives; notes.
September 8: Covariant derivative, affine connections; notes.
September 10: Fundamental theorem of Riemannian geometry, derivatives of tensors, Christoffel symbols; notes.
September 12: Homework 1 problem solving.
September 15: Christoffel symbols, definition of curvature; notes.
September 17: Symmetries of the curvature, Ricci, scalar curvature; notes.
September 22: The fundamental curvature equations and the equations of Riemannian geometry; notes.
September 24: Radius functions, radial curvature equations.
September 26: Examples; constant curvature manifolds, spheres, projective space.
September 29: Connections on vector bundles.
October 1: Curvature, Bianchi identities, ivariant polynomials.
October 3: Chern-Weil theory.