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, invariant polynomials.
October 3: Classification of invariant polyomials for matrix groups.
October 6: Chern-Weil theory.
October 8: Characteristic classes, general definition and properties.
October 10: Pontryagin class, properties.
October 13: Characteristic classes are topological invariants.
October 20: Complex vector bundles, hermitian vector bundles.
October 22: Chern class, Chern character via Chern--Weil theory.
October 24: Properties of the Chern class and Chern character.
October 27: Other definitions of Chern classes (Atiyah class).
October 29: Relationships among various characteristic classes.
October 31: Almost complex structures on vector spaces and manifolds.
November 3: Integrability, the ∂-operator.
November 5: Riemann surfaces and other examples of complex manifolds.
November 7: Holomorphic vector bundles, examples and properties.
November 10: Sheaf cohomology and Dolbeaults theorem.
November 12: The sheafy interpretation of characteristic classes (return the Atiyah class).
November 14: Back to hermitian vector spaces, fundamental form. Kähler manifolds.
November 17: Examples of Kähler manifolds. Properties of Riemann curvature for Kähler manifolds.
November 19: Lefshetz operator.
November 21: The sl(2) action on the cohomology of a Kähler manifold.
November 24: The ddc-lemma.
December 1: The fundamental theorem of elliptic complexes.
December 3: The Hodge decomposition of a compact Kähler mnaifold.
December 5: Hyperkähler manifolds.
December 8: Commutative dg algebras, minimal models, formality (definition).
December 10: Formality for compact Kähler manifolds.