Fall 2021: Math 232A Graduate Geometry 1
This course will focus on the role differential forms, the theory of bundles and connections play in modern geometry.
Monday, Wednesday in 268 Skye Hall at 2-3:20 (September 27 -- December 1)
Office hours: By appointment in person or on Zoom, or just stop by my office (232 Skye Hall) and we can chat if I'm free.
Some topics we will cover/review: Vector bundles, Lie derivatives, differential forms, the exterior derivative and Riemannian metrics. Along the way we will prove an important foundational result called the Frobenius theorem, define how to integrate on manifolds and prove the fundamental theorem of calculus (Stokes' theorem). We will also discussion the relation between differential forms and cohomology (de Rham's theorem), and the Hodge star (Hodge's theorem).
In the remainder of the course we will use the language of differential forms to talk about connections on vector bundles and the related concepts of parallel transport, holonomy and curvature. As a final goal, we either state Berger's classification of Riemannian holonomy groups or introduce the Yang-Mills and related equations (Self-duality, monopole, Kapustin-Witten,...).
Assessments: There will be 2 to 3 homework assignments and a final project.
Prerequisites: 205 B&C (or a good knowledge of the basics of manifolds and the fundamental group), a strong foundation in linear algebra will also be extremely helpful.
Main text: The geometry of differentials forms by S. Morita
Other useful texts:
Global Calculus by S. Ramanan
Differential forms and connections by R.W..R Darling
Differential geometry (Cartan's generalization of Klein's Erlangen program) by R.W. Sharpe
Introduction to smooth manifolds by J.M. Lee
Differential forms in algebraic topology by Bott and Tu
Homework:
Homework 1 due October 18
Homework 2 due November 3
Lecture notes: (Disclaimer: these lecture notes are probably full of errors)
Lecture 2: Tensor and exterior algebra
Lecture 3: Differential foms and the exterior derivative
Lecture 4: Exterior derivative and Lie derivative
Lecture 6: Lie derivative and distributions
Lecture 7: Frobenius theorem and differential ideals
Lecture 8: Proof of Forbenius theorem (integration next time)
Lecture 9: Integration on manifolds
Lecture 10: Stokes theorem
Lecture 11: Metrics on vector bundles and Riemannian metrics
Lecture 12: The Hodge star and inner products on forms
Lecture 13: The adjoint of d and the Laplace-Beltrami operator
Lecture 14: The Hodge theorem and Poincaré duality
Lecture 15: Proof of Hodge decomposition assuming regularity theorem and intro to connections
Lecture 16: Connections as operators, existence and the space of connections
Lecture 17: Local expressions and how they tranform: connection 1-forms
Lecture 18: Parallel transport and the 'horizontal distribution' of a connection
Lecture 19: Holonomy, flatness and representations of the fundamental group
Lecture 20: The Levi-Civita connection and Berger's classificiation result.