Math 535a: Differential Geometry, Spring 2022

Monday, Wednesday, Friday, 11:00-11:50am (time subject to change) in Zoom/KAP 137

Teaching Staff

Instructor Sheel Ganatra TA Tianle Liu
Office KAP 266D Office
e-mail sheel (dot) ganatra (at) usc (dot) edu (this is the best way to reach me) e-mail tianleli (at) usc (dot) edu

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).

Textbook and topics

The official course text is Introduction to Smooth Manifolds by John Lee (available online through USC libraries/SpringerLink). Additional references may 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 via Gradescope. You can handwrite or LaTeX your solutions (if handwriting, you will need to scan to PDF in order to upload to Gradescope; please let me know if you encounter any technical difficulties with this). 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, 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, 2/2 Homework 1.
Friday, 2/25 Homework 2.
Wednesday, 4/20 Homework 3.
Friday, 4/29 Homework 4.

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. 10 Welcome and overview of class. A review of point-set topology: topological spaces, metric spaces as an instance, and methods of constructing topological spaces (product and induced topologies). Hausdorff and separable topological spaces. Methods of constructing topological spaces (induced and quotient topologies). Homeomorphisms. Lee Appendix A, and any standard topology textbook, for instance Munkres. Notes
Jan. 12 Topological manifolds. Review of linear algebra. The notion of a category. Lee Appendix A and Chapter 1. The definition of a category is briefly mentioned at the end of Chapter 3. Notes
Jan. 14 Multivariable calculus review: differentiation and directional derivatives. k-times differentiable and smooth (= infinitely differentiable) functions. The inverse function theorem. The first definition of a smooth manifold. Lee Appendix A and Chapter 1. Notes
Jan. 17 No class (MLK day)
Jan. 19 Smooth manifolds. First examples of smooth manifolds. Lee chapter 1. Notes
Jan. 21 Examples of smooth manifolds continued. Comparing choices of smooth atlas (the notion of compatibility, an equivalence relation between atlases). Differentiable structures (which can be viewed as either an equivalence class of atlas or a choice of maximal atlas). Smooth functions from a (smooth) manifold to R. Lee chapters 1-2 Notes
Jan. 24 The algebra of smooth functions on a manifold. Operations on functions. Lee chapter 2. Notes
Jan. 26 A digression introducing manifolds-with-boundary, and the set of boundary points of such an object. Smooth maps between manifolds. Local coordinates. Lee chapters 1, 2. Notes
Jan. 28 The rank of a smooth map. Tangent vectors and the tangent space via parametrized curves (definition 1). Lee chapter 3 (see "Tangent vectors" and "Alternative definitions of tangent space") Notes
Jan. 31 Germs of smooth functions defined on a point, and the tangent space as the space of derivations on such germs. Lee chapter 3 (see "Tangent vectors and alternative definitions of tangent space") Notes
Feb. 2 The equivalence of two definitions of tangent space, and a third definition of tangent space. Lee chapter 3 (see "Tangent vectors and alternative definitions of tangent space") Notes
Feb. 4 The equivalence of the second and third definition of tangent space. The derivative of a smooth map at a point. Submanifolds. Lee Chapter 3 (see "Tangent vectors and alternative definitions of tangent space" and "The differential of a smooth map"), and Chapter 5 (the definition of submanifold we gave corresponds to "subsets that satisfy the local k-slice condition" in Lee) Notes
Feb. 7 Maps of constant rank, immersions, submersions. The rank theorem. Lee Chapter 4 Notes
Feb. 9 Finished proof of the rank theorem. Full rank is an open condition. Regular values and critical values. The preimage of a regular value is a smooth submanifold of the "correct" dimension. Lee Chapter 4,5 Notes
Feb. 11 Immersions and submanifolds. Lee Chapter 4,5 Notes
Feb. 14 Partitions of unity. Application to embedding manifolds into Euclidean space. Lee Chapter 4,5 Notes
Feb. 16 The tangent bundle of a manifold. Sard's theorem statement. Lee Chapter 3, 6 Notes
Feb. 18 Sard's theorem. Lee Chapter 6 Notes
Feb. 23 Finish proof of Sard's theorem and application to Whitney's embedding theorem. Cotangent spaces and the d operator. Lee Chapter 6, 11 Notes
Feb. 25 The cotangent bundle. One-forms and operations on one-forms. Lee Chapter 11 (note Lee calls one-forms "covector fields"). Notes
Feb. 28 Vector fields, and integrating vector fields. The fundamental theorem of flows (e.g., the fundamental theorem of ODEs). Lee Chapter 9, 10, Appendix D. Notes
Mar. 2 No class.
Mar. 4 The existence of global flows for a compact manifold. Some applications of flows. Derivatives of functions with respect to vector fields. The Lie bracket. Lee Chapter 9, 10 Notes
Mar. 7 The Lie bracket. Distributions. The Frobenius theorem. Lee Chapter 8, 9, 19 Notes
Mar. 9 Sketch of proof of the Frobenius theorem. Lie Groups (definition and examples) Lee Chapter 19, 7 Notes
Mar. 11 Vector bundles and sections. Examples. Restriction/pull back of vector bundles. Lee Chapter 10 Notes
Mar. 21 Vector bundles via gluing data. Constructing vector bundles from representations. Lee Chapter 10 Notes
Mar. 23 Operations on vector bundles coming from operations on vector spaces. The tensor product of vector spaces. Lee Chapter 10, 12 Notes
Mar. 25 The tensor and exterior algebras, and the s-fold wedge product of a vector space. Lee Chapter 12 Notes
Mar. 28 More about wedge products and the determinant. Tensor and wedge powers of vector bundles. Lee Chapter 12 Notes
Mar. 30 Differential k forms as sections of the wedge powers of the cotangent bundle. Exterior derivative. Lee Chapter 12, 14 Notes
Apr. 1 Exterior derivative continued. Integrating 1-forms. Lee Chapter 14, 11 (Line integrals) Notes
Apr. 4 Orientations and orientability Lee Chapter 15 Notes
Apr. 6 Orientations continued. De Rham cohomology. Lee Chapter 15, 17 Notes
Apr. 8 First computations in de Rham cohomology. Functoriality and homotopy invariance. Lee Chapter 17 Notes
Apr. 11 Homotopy invariance property for de Rham cohomology (statement), and application to Poincare Lemma (= de Rham cohomology of Euclidean spaces). The Mayer-Vietoris long-exact sequence (LES): statement and a sample computation. Lee Chapter 17 Notes
Apr. 13 Homological algebra: short exact sequences of cochain complexes and their associated long exact sequences of cohomology groups. Application of this to proving the Mayer-Vietoris LES. Lee Chapter 17 Notes
Apr. 15 Homological algebra continued. Chain homotopies between co-chain maps. The Lie derivative of a differential form or vector field along a vector field. Lee Chapter 17, 9 (Lie derivatives), 12 (Lie derivatives of more general tensors), 14 (Lie derivatives of forms). Notes
Apr. 18 Cartan's magic formula for computing Lie derivatives in terms of interior multiplication and exterior derivative. Application to proving homotopy invariance property in de Rham cohomology. Lee Chapter 14 Notes
Apr. 20 Additional examples in de Rham cohomology. Euler characteristic. Forms with compact support and compactly supported de Rham cohomology, and first computations of it. Functoriality properties of compactly supported de Rham cohomology. Lee Chapter 17. Notes
Apr. 22 Integration of (compactly supported) differentiable forms over oriented manifolds. Lee Chapter 16 Notes
Apr. 25 No class.
Apr. 27 Stokes' theorem and proof sketch. Lee Chapter 16 Notes
Apr. 27 (extra/bonus virtual lecture) Completed proof of Stokes' theorem. Proof of the theorem that integration induces an isomorphism between the top compactly supported cohomology of an oriented connected manifold and the real numbers. A statement of Poincare duality. Lee Chapter 16 and Chapter 17 (Theorem 17.30 and Lemma 17.27 in particular, though our treatment and statement of Poincare duality more follows Bott and Tu). Notes
Apr. 29 The top-dimensional compactly supported cohomology of a connected oriented manifold. Degree theory: The degree of a smooth map between connected, compact, oriented manifolds of the same dimension: Two definitions of degree (defined cohomologically and geometrically) and a proof they coincide. Properties/first computations of degree and some applications to studying homotopy classes of maps and vector fields on spheres. Lee Chapter 17 Notes