Math 535a: Differential Geometry, Spring 2017

Monday and Wednesday (and occasionally Friday), 10:30-11:50am in KAP 134

Teaching Staff

Instructor Sheel Ganatra TA Viktor Kleen
e-mail sheel (dot) ganatra (at) usc (dot) edu (this is the best way to reach me) e-mail kleen (at) usc (dot) edu
Telephone (213) 740-2417
Office Hours This week: by appointment. Office Hours

Course Description and Prerequisites

Math 535a gives an introduction to geometry and topology of smooth (or differentiable) manifolds and notions of calculus on them, for instance the theory of differential forms. We will assume familiarity with undergraduate topology, at the level of USC's Math 440 or equivalent. Exposure to theoretical linear algebra will also help (but will be quickly reviewed).

Announcements

Textbook and topics

The official course text is Foundations of Differential Manifolds and Lie Groups by Frank Warner. We will frequently deviate from this book; additional references will be posted on an ongoing basis.

Some topics to be covered include:

A more detailed lecture plan (updated on an ongoing basis, after each lecture) will be posted below.

Grading System

Homework Assignments

Homeworks will be posted here on an ongoing basis (roughly a week or more before they are due) and will be due on the date listed. You can handwrite or LaTeX your solutions. You may work with others and consult references (including the course textbook), but the homework you turn in must be written by you independently, in your own language, and you must cite your sources and collaborators.

Note: Homework deadline extensions are possible upon arrangement with me (with our TA, Viktor CC'ed) up to 3 times a semester, and at most once per HW. If you decide to request an extension from me, there will be no grading penalty if the HW is submitted within a couple days -- I'll set a more precise deadline each request. Beyond this, late homework will not be accepted.

Due date Assignment
Wednesday, 1/25 Homework 1. Solutions.
Friday, 2/3 Homework 2. Solutions.
Monday, 2/13 Homework 3. Solutions.
Friday, 2/24 Homework 4. Solutions.
Monday, 3/20 (Note extension!) Homework 5. Solutions.
Friday, 3/31 Homework 6. Solutions.
Wednesday, 4/19 Homework 7.
Friday, 4/28 Homework 8 (half weight assignment).

Note taking

I will ask each of you to be the official note taker for 4-5 lectures, and produce a set of typed (and revised) notes for the lecture. These should be completed and e-mailed to me within one week of the lecture, or one week of the day you were assigned the lecture, whichever is greater (so, if you were assigned on January 10 to write lecture notes for the January 9 class, you would have until January 17). You are required to use LaTeX to typeset these notes; now is a good time to learn LaTeX if you have not already. As a LaTeX template, please use the January 9th lecture notes, whose TeX (and PDF) are provided below.

Note: Some limited extensions are generally possible for note-taking assignments. Please e-mail me to arrange.

Takehome Final Exam

The takehome final will be a one-week exam, assigned via e-mail. You are strongly encouraged, but not required, to type up your solutions to the exam in LaTeX. When solving the problems, you can use any results we have used, stated, or proved in class or on your homeworks. If you wish you may consult other references (such as the textbook, coursenotes, homeworks, other references) for definitions, but you must solve these problems completely on your own. You may not discuss your solutions with other people except for the course staff (Sheel and Viktor).

Lecture Plan

Lecture topics by day will be posted on an ongoing basis below. Future topics are tentative and will be adjusted as necessary.

Day Lecture topics References and remarks Notes
Jan. 9 Welcome and overview of class. A review of point-set topology: topological spaces, metric spaces as an instance, and homeomorphisms. Any standard topology textbook, for instance Munkres. TeX. PDF
Jan. 11 Review: Linear algebra and multivariable calculus. The derivative linear map, partial/directional derivatives; higher order derivatives and differentiability, and smooth/C^infty functions. The notion of a category. Any standard theoretical linear algebra textbook, e.g., Axler, and a multivariable calculus book, e.g., Spivak.
Jan. 13 The chain rule. Diffeomorphisms of open sets in R^n. The inverse function theorem (statement). Topological and smooth manifolds.
Jan. 18 Examples of smooth manifolds. Equivalences of atlases, and differentiable structures (a.k.a. maximal atlases). Smooth functions on smooth manifolds.
Jan. 20 Smooth functions and smooth maps. The rank of a smooth map.
Jan. 23 Submersions and immersions, and prototypical examples. Regular values, critical values, and critical points. The implicit function theorem (for submersions), and applications to constructing manifold structures on preimages.
Jan. 25 Immersions, embeddings, and submanifolds. The implicit function theorem (for immersions). A first definition of tangent spaces (for submanifolds of R^n).
Jan. 30 Two intrinsic definitions of the tangent space to a manifold at a point: via equivalence classes of parametrized curves and derivations from the algebra of germs of smooth functions.
Feb. 1 A third definition of the tangent space to a manifold at a point. The derivative of a function as a map between tangent spaces. The tangent and cotangent bundles of a manifold.
Feb. 6 The cotangent space to a manifold at a point. The cotangent bundle, and functoriality with respect to smooth maps. One-forms and vector fields. Flows.
Feb. 8 Flows and the fundamental theorem of ODE. The action of vector fields on functions, and the Lie bracket of vector fields. Distributions and integral submanifolds. The Frobenius theorem.
Feb. 13 More on the Frobenius theorem. Partitions of unity, and an application to embedding manifolds in Euclidean space.
Feb. 15 Sard's theorem, and an application to Whitney's embedding theorem.
Feb. 20 No class (President's day).
Feb. 22 Lie groups and Lie group representations. Vector bundles and sub-bundles.
Feb. 24 (note Friday class)! More about vector bundles. Sections. Methods of constructing old vector bundles out of new vector bundles. Linear algebra: tensor products. Some notes from an old linear algebra course giving some formal properties of the tensor product (without complete proofs). See also Warner Sections 2.1-2.2.
Feb. 27 Linear algebra: more tensor products. The tensor algebra of a vector space. Wedge products and the exterior algebra of a vector space. Determinants. See here for some notes from an old linear algebra course giving some formal properties of the wedge product (without complete proofs). See also Warner Sections 2.4-2.6.
Mar. 1 Tensor product and wedge product of vector bundles. Differential forms (as sections of the kth exterior power bundle of the cotangent bundle). The exterior derivative. Integrating 1 forms.
Mar. 6 Orientability and orientations, for vector spaces and vector bundles. Examples. The classification of two manifolds (a statement of result).
Mar. 8 No class.
Mar. 13-17 Spring break (no class).
Mar. 20 De Rham cohomology. Functoriality for de Rham cohomology, and first examples.
Mar. 22 Statement of a crucial properties of de Rham cohomology: homotopy invariance. The Poincare lemma (de Rham cohomology of R^n). Exact sequences and the Mayer-Vietoris Sequence.
Mar. 24 A detour into homological algebra: co-chain complexes and co-chain maps. Short exact sequences of co-chain complexes. Long exact sequences in cohomology from short exact sequences of chain complexes. Introduction to the Lie derivative.
Mar. 27 More about Lie derivatives. Interior products, and Cartan's "magic" formula relating Lie derivatives to interior products. The proof of homotopy invariance of de Rham cohomology.
Mar. 29 More example computations of de Rham cohomology. The Euler characteristic of a manifold. A variant theory: de Rham cohomology with compact support. Beginning integration: a review of the Riemann integral.
Mar. 31 Riemann integrals and the change of variables formula. Integration of differential forms on oriented manifolds. An introduction to manifolds-with-boundary, an enlargement of the class of manifolds.
Apr. 3-7 No class.
Apr. 10 More manifolds with boundary, and induced orientations. Stokes' theorem and some applications.
Apr. 12 An application of Stokes' theorem to the divergence theorem. Integration on cohomology, and a statement of Poincare duality. The top dimensional (compactly supported) cohomology of a manifold, and the cohomological degree of a map between compact manifolds.
Apr. 14 The degree of a map between (compact oriented) manifolds (without boundary). Cohomological and topological degree coincide. (Along the way: proof of the fact that top dimensional cohomology is one-dimensional.) An introduction to intersection theory.
Apr. 17 Intersection theory. Oriented zero-manifolds, intersections of transverse sub-manifolds, and intersections of transverse manifolds along maps (also known as "pre-images of submanifolds").
Apr. 19 The cohomological interpretation of intersection theory (assuming Poincare duality). A statement of the tubular neighborhood theorem. Orthogonal structures on vector bundles.
Apr. 19 Riemannian manifolds (manifolds with orthogonal structures on their tangent bundles). Proof of the tubular neighborhood theorem. Connections on vector bundles.
Apr. 24 More connections on vector bundles. (Local and global) covariantly constant frames for connections, and flat connections (connections which admit a local covariant constant frame near each point). Existence of connections. The space of connections as an affine space. The curvature of a connection. Local expression of a connection as differentiation plus a matrix of one-forms, and an interpretation of curvature in that regard.
Apr. 26 The Levi-Civita connection on a Riemannian manifold. Applications of Riemannian geometry, and a first relationship between the Riemannian geometry of a surface and its topology: the Gauss-Bonnet theorem.