Math 535a: Differential Geometry, Spring 2018

Monday, Wednesday, Friday, 11:00-11:50am (time subject to change) in KAP 134

Teaching Staff

Instructor Sheel Ganatra TA Dong Zhang
Office KAP 266D Office
e-mail sheel (dot) ganatra (at) usc (dot) edu (this is the best way to reach me) e-mail dongz (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).

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, 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, 1/17 Homework 1.
Monday, 1/29 Homework 2.
Friday, 2/9 (extended to Wednesday 2/14 for all students) Homework 3.
Monday, 2/26 Homework 4.
Monday, 3/9 Homework 5.
Friday, 4/13 Homework 6.
Friday, 4/27 Homework 7 (1.5x weight).

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. 8 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. Any standard topology textbook, for instance Munkres.
Jan. 10 More topology: methods of constructing topological spaces (induced and quotient topologies). Homeomorphisms. Second-countability and the notion of a topological manifold. The notion of a category. Any standard topology book, for instance Munkres.
Jan. 12 Review: Linear algebra, the notion of a category, and multivariable differential calculus: The derivative linear map and directional derivatives. partial/directional derivatives; higher order derivatives and differentiability, and smooth/C^infty functions. Any standard theoretical linear algebra textbook, e.g., Axler, and a multivariable calculus book, e.g., Spivak.
Jan. 15 No class (MLK day)
Jan. 17 The derivative linear map and directional derivatives. partial/directional derivatives; higher order derivatives and differentiability, and smooth/C^infty functions. The chain rule. Any standard theoretical linear algebra textbook, e.g., Axler, and a multivariable calculus book, e.g., Spivak.
Jan. 19 The chain rule. Diffeomorphisms of open sets in R^n. The inverse function theorem (statement). Topological and smooth manifolds.
Jan. 22 Smooth manifolds and examples of smooth manifolds: R^n, open subsets of smooth manifolds, products of smooth manifolds. S^1 and real projective space as other examples.
Jan. 23 (makeup class) A further example: the two-torus. Choice of atlas, and the notion of a maximal atlas or differentiable structure.
Jan. 24 Smooth functions on a (smooth) manifold. The notion of being a smooth function only depends on the differentiable structure (i.e., maximal atlas), not the specific atlas. Smooth bump functions, as a technical tool for the next class.
Jan. 25 (makeup class) Two smooth atlases on a manifold M are compatible/lie in the same differentiable structure if and only if their sets of C-infinity functions coincide (proof continued). Pulling back continuous functions, and the notion of equivalent smooth (or differentiable) structures.
Jan. 26 Smooth maps between manifolds, and diffeomorphisms. The rank of a smooth map.
Jan. 29 The rank of a smooth map continued. Submersions and immersions. The implicit function theorem.
Jan. 31 Critical values and regular values of a smooth map f: M --> N. The preimage of any regular value can be given the structure of a manifold (in fact a submanifold--a notion we'll define next class, or on homework).
Feb. 2 No class.
Feb. 5 No class.
Feb. 7 No class.
Feb. 9 Immersions and embeddings (which require the notion of a proper map).
Feb. 12 More about embeddings. and the notion of a submanifold. The notion of a tangent vector for submanifolds of R^N (two extrinsic definitions).
Feb. 14 Abstract (or intrinsic) notions of tangent space: (i) as equivalence classes of smooth curves.
Feb. 16 Abstract (or intrinsic) notions of tangent space: (ii) as derivations.
Feb. 21 Abstract notions of tangent space: (ii) as derivations (continued), and (iii) as the dual to the cotangent space (time permitting).
Feb. 23 Abstract notions of tangent space: (iii) as the dual to the cotangent space. The derivative of a differentiable map, as a map between tangent spaces.
Feb. 26 The tangent bundle.
Feb. 28 The cotangent bundle, and one-forms.
Mar. 2 One-forms and vector fields. Flows.
Mar. 5 More about flows. Flows define vector fields and vector fields determine at least local short-time flows (for compact manifolds, vector fields actually define flows). Vector fields act on functions, and the Lie bracket of vector fields.
Mar. 7 Vector bundles and sub-bundles. Distributions. The Frobenius theorem: a distribution is integrable if and only if it is involutive.
Mar. 9 A vector field which is non-zero at a point p can always locally be written as d/dx_1 for some coordinate system x_1, ... x_m near p. (note that this implies a special case of the Frobenius theorem).
Mar. 12-16 No class (spring break).
Mar. 19 Sard's theorem (first statement). More about the Frobenius theorem, and a sketch of proof.
Mar. 21 Partitions of unity and embedding manifolds in Euclidean space.
Mar. 23 Embedding manifolds into Euclidean space (continued). Whitney's theorem of embedding manifolds into small-dimensional Euclidean spaces, which is an application of Sard's theorem.
Mar. 26 (tentative) Sard's theorem and a proof.
Mar. 28 No class.
Mar. 30 More linear algebra: tensor products and their universal property.
April 2 The tensor algebra and exterior algebra. The k-fold wedge product of a vector space.
April 3 Makeup class in GFS 221. More about the k-fold wedge product of a vector space. The determinant.
April 4 Tensor and wedge products of the tangent and cotangent bundle. The k-th wedge power of the cotangent bundle, and an introduction to differential k-forms (which are sections of the k-th wedge of the cotangent bundle).
April 6 Operations on differential forms: wedging them, pullback, and exterior derivative.
April 9 The exterior derivative operator reviewed, and the definition of (kth) de Rham cohomology of a manifold (for each k). Integrating 1-forms.
April 11 Orientations on vector spaces, and the notion of an orientation-preserving map. If V is an oriented vector space, then a map T: V to V is orientation preserving iff det(T) is positive. The notion of orientability of a vector bundle (and an example of a non-orientable vector bundle). A manifold is orientable if its tangent (or equivalently cotangent) bundle is orientable.
April 12 Makeup class in GFS 221. Equivalent notions of a manifold being orientable. The (geometric) degree of a smooth map between compact oriented manifolds.
April 13 The Riemann integral, and integrating k-forms. Manifolds-with-boundary.
April 16 More manifolds-with-boundary. Boundary orientations, and Stokes' theorem.
April 17 Makeup class in GFS 221. A return to de Rham cohomology: closed and exact forms. Computations of the de Rham cohomology of a point, R, and S^1. Some general facts about de Rham cohomology, including the bottom cohomology, and the top cohomology of a compact oriented manifold. Using integration to detect cohomology classes.
April 18 More properties of De Rham cohomology: functoriality and homotopy invariance. The de Rham cohomology of R^n. Introduction to the Mayer-Vietoris sequence.
April 19 Makeup class in GFS 221. More about the Mayer-Vietoris sequence, and using it to re-compute the cohomology of S^1.
April 20 The degree of a smooth map between connected, compact, oriented manifolds of the same dimension: Two definitions of degree (defined geometrically and cohomologically) and a proof they coincide. Properties of degree and some applications to studying homotopy classes of maps and vector fields on spheres.
April 23-27 No class (though brief meeting April 23 to fill out Course Evaluations)..