Math 535b: Advanced Differential Geometry, Spring 2020

Monday, Wednesday, Friday, 11:00-11:50am in KAP 134

Teaching Staff

Instructor	Sheel Ganatra
Office	KAP 266D
E-mail	sheel (dot) ganatra (at) usc (dot) edu (this is the best way to reach me)
Office hours (virtual)	Mon 3:15-4:15pm, Wed 10-11am, Thurs 3-4pm or at other times by Zoom appointment.

Announcements

3/9/2020: Our class will move to (remote) online instruction via Zoom for the remainder of the semester (as a campus-wide measure in response to the COVID-19 pandemic). Please let me know ASAP if you are not receiving Zoom class meeting invitations through Blackboard.
2/14/2020: There will be no class the week of 2/17-2/21; we will resume class on Monday 2/24.

Course Description and Prerequisites

Math 535b is an advanced course on the geometry of (real and complex) manifolds. The focus in Spring 2020 is on the geometry of manifolds equipped with extra structure, particularly of the following three varieties (with an emphasis on the latter two):

Riemannian geometry (the study of manifolds with Riemannian structure, i.e., a metric induced by a family of inner products on tangent spaces).
Kähler geometry (the study of complex manifolds with a Kähler metric, which is a special type of (Hermitian) metric whose purely imaginary part is closed).
Symplectic geometry (the study of real even-dimensional manifolds with a symplectic form, which is a non-degenerate closed 2-form).

Kähler manifolds are in particular complex manifolds which are Riemannian and symplectic in a compatible way. We will discuss examples of all three types of manifolds and develop a basic toolkit for studying geometry in each of these settings. An emphasis of the course will be on understanding rigidity properties of the latter two geometries: ways in which their geometry and topology (as measured by e.g., their cohomology groups, or by properties of certain maps or vector fields) are tightly constrained in comparison to manifolds that do not possess these structures.

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

In the last roughly two weeks of the course, the class will become student run: students will give a series of talks about advanced topics in one or more of these areas. A list of potential topics will be posted at a slightly later date.

We will assume basic familiarity with the geometry of manifolds, at the level of USC's Math 535a course or equivalent.

References

Some planned topics to be surveyed include:

Some aspects of manifold theory: vector bundles, tubular neighborhoods, Poincare duality, and intersection theory for submanifolds.
Connections and curvature (on vector bundles), and an introduction to Riemannian geometry: Riemannian metrics and their associated Levi-Civita connections, parallel transport, geodesics. (time permitting) Applications of Riemannian geometry and the interplay with topology: the tubular neighborhood theorem, Gauss-Bonnet formula, gradient vector fields and introduction to Morse theory.
Symplectic geometry: Symplectic forms, Lagrangian submanifolds, Hamiltonian vector fields and symplectomorphisms, Darboux's theorem and the Weinstein tubular neighborhood theorem, Moser's theorem. Constructions of symplectic manifolds (via fibrations, blow up, and cut and paste). Compatible almost-complex structures. Arnold's conjecture and (time permitting) an introduction to Floer homology.
Complex manifolds and ''almost-complex manifolds'' (manifolds equipped with almost-complex structures), and their relationship via the Newlander-Nirenburg theorem.
Kähler geometry: Kähler forms, and the Hodge decomposition for (the de Rham cohomology of) Kähler manifolds. Examples coming from complex projective space and smooth projective complex varieties. The Kodaira embedding theorem.

There is no official course textbook, but we will frequently refer to the following books (and likely others), primarily Lee and Cannas da Silva:

Raoul Bott and Loring Tu, Differential forms in Algebraic Topology (Specifically Chapter 1, which gives a nice treatment of De Rham cohomology, Poincaré duality using differential forms, the Künneth theorem, vector bundles, ...).
John Lee, Riemannian manifolds: An Introduction to Curvature .
Ana Cannas da Silva, Lectures on symplectic geometry, available online.
Raymond Wells, Differential analysis on complex manifolds.
Claire Voisin, Hodge Theory and Algebraic Geometry, I.

We will also make key use of the following course notes:

Professor Ko Honda's Differential geometry course notes: first semester and second semester.
Professor Denis Auroux's 2007 MIT course, 18.966: Geometry of Manifolds
Professor Ralph Cohen's lecture notes, The topology of fiber bundles (we will use this as a supplementary reference at times when we might discuss vector and or principal bundles).

For review of the introductory geometry and topology of manifolds, you may wish to consult the textbook used frequently in Math 535a, Foundations of Differential Manifolds and Lie Groups by Frank Warner. An alternative introductory textbook is Introduction to Smooth Manifolds by John Lee. A nice complementary reference is Guillemin and Pollack's Differential Topology, particularly its chapter on intersection theory.

Grading System

Your grade in the course, a measure of your demonstrated understanding of course topics, will come from participation along with two (individual) final assignments:

A lecture given in the final weeks of the course, on an advanced topic building on what is covered in the class. and
A final expository paper, at least 5 pages long, about an advanced topic as above. This is allowed, but does not need to exactly coincide with your lecture topic. In fact, even if the topic is the same, the paper should necessarily have more details and/or differing emphasis from what is lectured about in class.

A list of possible topics for the above assignments will be provided later in the semester. You are also welcome to propose a different topic, subject to instructor approval.

Homework Assignments

Some optional (ungraded) homeworks exercises may be assigned periodically throughout the semester. They will be posted here on an ongoing basis.

Date	Exercises	Notes
1/17	Problems 1-4 of this HW assignment on vector bundles	These exercises give various equivalent ways to think about vector bundles and also justify the various operations we perform on vector bundles. (Warning: problem 3 is fairly long, and requires some understanding of the definition of a category)

Student lectures and final papers

Details about your final lecture and paper assignment are now available here.

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
Jan. 13	Lecture 1: Welcome + an overview of the class. Review: Manifolds and vector bundles, and the tangent and cotangent bundles.
Jan. 15	Lecture 2: More overview of the class. Review: manifolds and vector bundles. Vector sub-bundles and operations on vector bundles (quotients, tensor products, wedge products, etc.). The normal bundle of a submanifold. Vector fields and differential forms.
Jan. 17	Lecture 3: Local (and global) frames of vector bundles. Vector fields and differential forms. Wedge product and exterior derivative, and de Rham cohomology.	Auroux's lecture 1 and Math 535a.
Jan. 20	No class (MLK day)
Jan. 22	Lecture 4: Operation involving forms and vector fields: contraction, Lie derivative (and flows), and basic properties. Compactly supported cohomology.
Jan. 24	Lecture 5: Orientability and orientations, and integration of forms (review). The top compactly supported cohomology of a manifold M is rank 1 (if M is connected), with an isomorphism to R given by integration. Wedge products and the Poincare duality pairing.
Jan. 27	Lecture 6: Cohomological and geometric intersection numbers of submanifolds, and the theorem that they agree.
Jan. 29	Lecture 7: A sketch of proof that cohomological = geometric intersection numbers (which takes a detour through the tubular neighborhood theorem and Thom forms). The tensoriality Lemma: Maps between spaces of sections come from maps of vector bundles iff they are C-infinity-linear, i.e., tensorial.	Bott and Tu Chapter 1 (for Thom forms).
Jan. 31	Lecture 8: Proof of the tensoriality Lemma. Inner products on vector spaces and orthogonal structures on vector bundles. Orthogonal structures and Riemannian metrics (= orthogonal structures on the tangent bundle). Orthogonal structures always exist (by a partition of unity argument). Orthogonal structures can be restricted to sub-bundles and used to construct orthogonally complementary bundles.	Honda Lecture 37 (for Riemannian metric, orthogonal structure is a generalization to the case of vector bundles).
Feb. 3	Lecture 9: Examples of orthogonal structures and Riemannian metrics: Euclidean space, submanifolds inherit metrics (and their normal bundles inherit orthogonal structures). The associated metric on a Riemannian manifold (where the distance from p to q is the infimum arc length of any piecewise smooth path from p to q).	Lee Chapter 3, Chapter 6.
Feb. 5	Lecture 10: Finished the proof that the infimum over arc length is a metric on any Riemannian manifold. Riemannian metrics induce an identification between vector fields and one-forms, and hence we can define the gradient vector field of any smooth function f (as the metric dual of df).
Feb. 7	Lecture 11: Connections on vector bundles (definitions, along with motivation coming from the notion of a geodesic and/or the desire to take directional derivatives). Covariantly constant frames (global and local) and flat connections.
Feb. 10	Lecture 12: Connections on E as an affine space over Omega1(End(E)) (1-forms with values in endomorphisms of E). Connections in local coordinates as d + A. Connections on the tangent bundle and the associated Christoffel symbols (obtained by further expanding the entries of A in local coordinates).	Honda lectures 35-36, Auroux lecture 9.
Feb. 12	Lecture 13: Covariant derivatives of sections along curves. Parallel transport along a curve induces an isomorphism between the fibers of the vector space at the the endpoints.	Lee Chapter 4 (the section on parallel transport, written for connections on the tangent bundle, generalizes immediately to other vector bundles)
Feb. 14	Lecture 14: Pullback connections. Parallel transport for flat connections only depends on the homotopy class of the curve. The monodromy (or holonomy representation) associated to a flat connection. The curvature of a general connection, and the proof that it vanishes if the connection is flat.	Lee Chapter 4, Honda lectures 35-36, Auroux lecture 9.
Feb. 17-21	No class.
Feb. 24	Lecture 15: The curvature of a connection vanishes if and only if the connection is flat. Geodesics, and their existence and uniqueness.	Lee Chapter 4
Feb. 26	Lecture 16: More about the existence and uniqueness of geodesics. A local expression for curvature and the interpretation as the obstruction preventing a certain operator to give a chain complex. When the connection is flat, we do get a chain complex computing the (twisted) de Rham cohomology with coefficients in the local system (or flat connection).	Lee Chapter 4.
Feb. 28	Lecture 17: Connections that are compatible with a metric (or orthogonal) structure on a vector bundle. The Levi-Civita connection on a Riemannian manifold (the unique connection which is compatible with the metric and torsion free). The exponential map.	Lee Chapter 5 (specifically see Theorem 5.10).
Mar. 2	Lecture 18: More about the exponential map. An application: the tubular neighborhood theorem.	Lee Chapter 5
Mar. 4	Lecture 19: The proof of the tubular neighborhood theorem.	Lee Chapter 5, Auroux Lecture 5 (another sketch of proof of tubular neighborhood theorem)
Mar. 6	Lecture 20: Symplectic vector spaces.	Cannas da Silva Chapter 1, Auroux Lecture 2
Mar. 9	Lecture 21: Special linear subspaces of symplectic vector spaces: symplectic, isotropic, coisotropic, and Lagrangian subspaces. The symplectic volume element associated to a symplectic structure. The symplectic linear group. Symplectic manifolds.	Cannas da Silva Chapter 1, Auroux Lecture 2
Mar. 11	Lecture 22: Examples of symplectic manifolds: Euclidean 2n-dimensional space, quotients by group actions which preserve the symplectic structure, orientable 2-manifolds equipped with area form, products of symplectic manifolds, cotangent bundles.	Cannas da Silva Chapter 1-2, Auroux Lecture 2-3.
Mar. 13	Lecture 23: More about the symplectic structure on the cotangent bundle. Symplectic, Lagrangian, isotropic, and coisotropic submanifolds of symplectic manifolds.	Cannas da Silva Chapter 2-3. Auroux Lecture 3.
Mar. 16-20	No lecture (spring break).
Mar. 23	Lecture 24: Examples of Lagrangian submanifolds: conormal bundles in cotangent bundles (including the zero section and cotangent fibre as extreme possibilities), graphs of symplectomorphisms. Hamiltonian vector fields.	Cannas da Silva Chapter 3, 18.
Mar. 25	Lecture 25: Isotopies and flows. Flows of Hamiltonian vector fields are symplectomorphisms. Moser's theorem.	Cannas da Silva Chapter 6.1, 18, 7
Mar. 27	Lecture 26: Proof of Moser's theorem. The relative Moser theorem. Darboux's theorem.	Cannas da Silva Chapter 7-8.
Mar. 30	Lecture 27: Some linear algebra: (Linear) complex structures. Compatibility of (linear) complex structures with a given (linear) symplectic structure.	Cannas da Silva Chapter 12.2
Apr. 1	Lecture 28: Compatible triples on a vector space and the relationship with Hermitian inner products. A map which is two out of three of (symplectic, orthogonal, complex linear) is all three and unitary.	Cannas da Silva Chapter 12-13
Apr. 3	Lecture 29: Compatible almost complex structures on a symplectic manifold.	Cannas da Silva Chapter 12-13
Apr. 6	Lecture 30: Weinstein's Lagrangian neighborhood theorem.	Cannas da Silva Chapter 8.3
Apr. 8	Lecture 31: Proof of the Lagrangian neighborhood theorem continued. The complexification of a real vector space, and the (1,0) and (0,1) decompositions of this complexification associated to a (linear) complex structure on the underlying real vector space.	Cannas da Silva Chapter 8.3, 14.
Apr. 10	Lecture 32: The complexified tangent bundle and its (1,0) and (0,1) splittings (associated to an almost complex structure). Differential forms of type (p,q).	Cannas da Silva Chapter 14.
Apr. 13	Lecture 33: J-holomorphic maps between almost complex manifolds. Complex manifolds.	Cannas da Silva Chapter 14-15.
Apr. 15	Lecture 34: Integrable almost complex structures (those induced by the structure of a complex manifold). The del and del-bar operators, and (p,q) Dolbeault cohomology (defined for a complex manifold because d = del + del-bar on complex manifolds).	Cannas da Silva Chapter 15.
Apr. 17	Lecture 35: The Nijenhuis tensor and the Newlander-Nirenberg theorem (which characterizes integrable almost complex structures as those whose Nijenhuis tensor vanishes or satisfy some equivalent criterion such as d = del + del-bar).	Cannas da Silva Chapter 14-15.
Apr. 20	Lecture 36: Kahler manifolds and Kahler forms. Kahler forms are (1,1) and can be obtained from strictly plurisubarhmonic (spsh) functions.	Cannas da Silva Chapter 16.
Apr. 22	Lecture 37: Kahler potentials (spsh functions inducing a given Kahler form) always exist locally. The Fubini-Study Kahler form on complex projective space.	Cannas da Silva Chapter 16 (and homework 10).
Apr. 24	Lecture 38: Examples of Kahler manifolds as submanifolds of projective space or complex Euclidean space. Complex projective and affine varieties. The Kodaira embedding theorem (statement), and Stein manifolds (a characterization of which necessarily non-compact complex manifolds embed into complex Euclidean space). Statement of the Hodge decomposition and immediate corollaries.	Cannas da Silva Chapter 17.
Apr. 27	Lecture 39: Overview of Hodge theory and the Laplacian on a compact Riemannian manifold.	Cannas da Silva Chapter 17.
Apr. 29	Lecture 40: Overview of Hodge theory and various Laplacians (and their relationships) on a compact Kahler manifold. The Hodge diamond, and the hard Lefschetz theorem. The Andreotti-Frankel/Milnor theorem that Stein manifolds have the topology of a half-dimensional CW complex.	Cannas da Silva Chapter 17.
May 1	Lecture 41: Sketch of proof of the Andreotti-Frankel/Milnor theorem (involving an introduction to Morse theory).