Math 535b: Advanced Differential Geometry, Spring 2019

Monday, Wednesday, Friday, 11:00-11:50am (time subject to change) 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 This week (3/25-3/29): Monday 2-3pm, Wednesday 2-3pm, Friday 12-1pm.


Course Description and Prerequisites

Math 535b is an advanced course on the geometry of (real and complex) manifolds. The focus in Spring 2019 is on the geometry of manifolds equipped with extra structure, particularly of the following three varieties (with an emphasis on the latter two):

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 two or three 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.


Some planned topics to be surveyed include:

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

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

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 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 will be assigned periodically throughout the semester. They will be posted here on an ongoing basis.

Date Assignment Notes

Student lectures and final papers

Some 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 Notes
Jan. 7-11 No class. First class will meet Monday Jan. 14.
Jan. 14 Welcome + an overview of the class.
Jan. 16 Review: manifolds and vector bundles. The tangent and cotangent bundle (and exterior powers of the latter). Differential forms and vector fields, and operations between them (exterior derivative, wedge product, contraction, Lie derivative + Cartan's magic formula). De Rham cohomology. Auroux's lecture 1 and Math 535a.
Jan. 18 Compactly supported cohomology, as the cohomology of compactly supported differential forms. Some properties of compactly supported cohomology (contravariant functoriality proper maps, covariant functoriality for open inclusions), and the compactly supported cohomology of R. Poincaré duality on M as a perfect pairing between cohomology in degree k and compactly supported cohomology in degree dim(M)-k (which is just cohomology if M is compact). The Mayer-Vietoris sequence. Auroux's lecture 1 + Bott and Tu Chapter 1.
Jan. 21 No class (University holiday)
Jan. 23 The Poincaré dual class of a compact oriented submanifold. Intersection numbers of compact oriented (complementary-dimensional) submanifolds, defined cohomologically and geometrically (when the submanifolds intersect transversely), and the theorem that they agree. Statement of the tubular neighborhood theorem (and the fact that tubular neighborhoods can be used to define Thom forms associated to compact submanifolds, which we didn't define but which are particularly nice representatives of the Poincaré dual classes). Bott and Tu Chapter 1.
Jan. 25 More intersection theory: isotopy invariance of intersection numbers, disjoinability of submanifolds (and intersection numbers as an obstruction to disjoinability), and the general position/transversality theorem (which allows one to define self-intersection numbers). Changing topics: Orthogonal structures on vector bundles, and Riemannian metric (= orthogonal structure on the tangent bundle). Orthogonal structures always exist (by a partition of unity argument). Guillemin and Pollack Chapters 2-3 (for transversality theory), . Honda Lecture 37 (for Riemannian metric, orthogonal structure is a generalization to the case of vector bundles).
Jan. 28 Connections on vector bundles. The connection associated to a trivialization. Flat connections, the connections which admit (locally) covariantly constant frames. Connections always exist, and the expression of connections locally in terms of d (the flat connection in local coordinates) plus a connection 1-form (a matrix of 1-forms) A. Honda Lectures 34-35.
Jan. 30 No class.
Feb. 1 No class.
Feb. 4 Connections on E as an affine space over Omega1(End(E)) (1-forms with values in endomorphisms of E). A digression on what it means for maps from sections to sections to be tensorial (i.e., C-infinity(M) linear). Metric connections (connections which are compatible with a metric). The curvature of a connection, as a tensorial operation on a pair of vector fields (anti-symmetric in the pair) and a section. Honda lectures 35-36, Auroux lecture 9.
Feb. 6 A connection is flat iff its curvature vanishes. Curvature in local coordinates (dA + A wedge A), and thought of as an obstruction to (d + A) squaring to zero (and inducing a chain complex).
Feb. 8 No class.
Feb. 11 Change of frame, and its effect on local expressions for the connection and curvature. Connections on the tangent bundle and their torsion. Theorem: there exists, a unique connection, called the Levi-Civita connection, which is both compatible with a given Riemannian metric and also torsion-free. Honda Lectures 37 and 40, Auroux Lecture 9
Feb. 13 (morning) Examples of Riemannian manifolds: R^n with its standard metric, and any submanifold of a Riemannian manifold inherits a Riemannian structure. An example of a vector bundle with a metric: the tanget bundle to an ambient Riemannian manifold (restricted along a submanifold), and/or the normal bundle of a submanifold of a larger manifold. Isometric embeddings, and isometries. Proof of the theorem of existence of the Levi-Civita connection. The Christoffel symbols associated to a connection on the tangent bundle (in any set of local coordinates). Lee Chapter 5 (specifically Theorem 5.4), Honda Lecture 37 There are many examples of Riemannian manifolds in Lee's book; Lee's proof of Theorem 5.4 derives an explicit formula for the Christoffel symbols of the Levi-Civita connection in terms of the local coordinate expansion of the metric.
Feb. 13 (afternoon make-up class) Connections on vector bundles and the induced covariant derivative of sections of E (i.e., vector fields if E = TM) along curves in M. Parallel transport along paths (using the connection or equivalently covariant derivative). The distance between two points on a Riemannian manifold as the infimum length of any path. If a path realizes the infimum, then (up to reparametrization) it can be realized by a geodesics: these are more generally "constant acceleration" paths (defined using covariant derivative); they also are critical points of the energy functional. The existence and uniqueness theorem of (at least short-time) geodesics, and the exponential map. An application of the exponential map to proving the tubular neighborhood theorem. Lee Chapter 4 (for covariant derivative, parallel transport, geodesics), Lee Chapter 5 (for exponential map), Auroux Lecture 5 (for sketch of proof of tubular neighborhood theorem).
Feb. 15 Small geodesics minimize distance, and in particular the infimum of lengths of paths from p to q defines a metric on (M,g) (whose topology coincides with the usual topology on M, though we didn't say this in class-exercise). The Gauss-Bonnet theorem: a relationship between curvature and topology for a 2-dimensional manifold. Lee Chapter 6 (for more details about the proof that small geodesics are distance minimizers).
Feb. 18 No class (University Holiday)
Feb. 20 A sketch of proof of the Gauss-Bonnet theorem. Key ingredients include: (a) the relationship between curvature of a surface in R3 (with its induced metric) and the degree of the associated Gauss map (via Gauss's Theorema Egregium, relating the second fundamental form to the curvature tensor), (b) an appeal to degree theory, and (c) the fact that for any manifold M, the intersection number of the zero section with itself in TM equals the intersection number of the diagonal with itself in M x M equals the Euler characteristic.
Feb. 20 (afternoon make-up class) Introduction to symplectic geometry. First, symplectic linear algebra: symplectic vector spaces and different types of subspaces they can have: symplectic subspaces, isotropic subspaces, coisotropic subspaces, and Lagrangian subspaces (= isotropic + coisotropic, or isotropic + maximal dimension). Linear symplectomorphisms. Any symplectic vector space has a standard basis and hence is linear symplectomorphic to R2n with its standard symplectic structure. Then, symplectic manifolds: first definition and examples: R2n, cotangent bundles, orientable 2-manifolds, products. Symplectomorphisms. Cannas da Silva Chapters 1 and 2 + Auroux lecture 2.
Feb. 22 The cotangent bundle example continued. Examples of symplectomorphisms. Symplectic, isotropic, coisotropic and Lagrangian submanifolds. Examples of Lagrangian submanifolds (for instance, the graph of a symplectomorphism, or a conormal bundle, such as the zero section or cotangent fibre, in a cotangent bundle. Also, graphs of closed 1-forms). Exact symplectic manifolds and their exact Lagrangian submanifolds. Cannas da Silva Chapters 2 and 3 + Auroux lecture 3.
Feb. 25 A reminder on flows (induced by vector fields) and isotopies (induced by possible time-dependent vector fields). The Hamiltonian vector field associated to functions on symplectic manifolds (which are called Hamiltonians by convention) (possibly time dependent, in which case the vector field is as well). Hamiltonian diffeomorphisms (the time-1 map associated to a flow/isotopy coming from a Hamiltonian vector field), and the proof that Hamiltonian diffeomorphisms are in particular symplectomorphisms. The Hamiltonian vector field associated to a Hamiltonian H is tangent to the level sets of H (i.e., the flow preserves H, unlike the gradient vector field of a function f with respect to a metric, whose flow increases f). Examples. (in particular, the example of H = kinetic + potential energy; the associated integral curves of X_H solve Hamilton's equations of motion, and the fact that H is preserved is the familiar conservation of energy) Cannas da Silva Sections 18.1-18.2, Auroux lectures 3 and 4.
Feb. 27 Arnold's conjecture on the number of fixed points of a (non-degenerate) Hamiltonian diffeomorphism. (this is an instance of rigidity of these diffeomorphisms; in the typical case for diffeomorphisms isotopic to id, the minimum number of fixed points is typically lower as given by the Lefschetz fixed point theorem). Examples. Hamiltonian diffeomorphisms versus symplectomorphisms in general. The flux of a symplectomorphism which is not Hamiltonian. Auroux lecture 4, Cannas da Silva Section 9.4
Feb. 27 (afternoon make-up class) Moser's theorem and the Moser trick. An application to the Darboux theorem. The local Moser theorem and an application to Weinstein's Lagrangian neighborhood theorem. Auroux lecture 5, Cannas da Silva Chapter 7; our treatment of the Weinstein neighborhood theorem is slightly closer to the treatment in McDuff-Salamon's "Introduction to symplectic topology"
Mar. 1 More about the Weinstein's Lagrangian neighborhood theorem. Complex structures on vector spaces. Same as last class and Cannas da Silva Chapter 12
Mar. 4 Complex structures J on a vector space which are compatible with a given symplectic structure Omega (and hence induce a metric g_J = Omega(-,J-)). Compatible complex structures exist: moreoever given a fixed Omega and any inner product g, there is a canonical almost Omega-compatible almost complex structure J one can build. (g is not necessarily equal to the resulting inner product g_J = Omega(-,J-)); instead one realizes g always equals Omega(-,A-) for some A which need not be a complex structure, and then apply polar decomposition to A to produce the desired J. Compatible triples are triples of the form (Omega, J, g_J) where J is compatible with Omega and g_J = Omega(-,J-) is the resulting inner product. Any two of Omega, J, g in a compatible triple determine the third. Auroux lecture 7 and Cannas da Silva Chapter 12 + 13.
Mar. 6 Examples of isotropic and coisotropic submanifolds. More about compatible triples. Almost complex structures on manifolds, and compatible (with a symplectic form) almost complex structures. Compatible (with a given omega) almost complex structures always exist and the space of such compatible structures is compatible. The space of symplectic structures compatible with a given almost complex structure is convex (hence contractible), although could be empty. Auroux lectures 7-8, Cannas da Silva Chapter 12 + 13.
Mar. 8 Complex vector bundles and almost complex submanifolds. Given a compatible (with omega) almost complex structure, almost complex submanifolds are symplectic submanifolds too. Hermitian inner products. Complex-valued connections (on complex bundles) and the first Chern class of a complex line bundle. Auroux lectures 8 + 10. The material about Hermitian inner products can be found in a linear algebra reference, such as Axler's "Linear Algebra done right" Chapter 6.
Mar. 11-15 No class (spring break).
Mar. 18 Complex structures and complexification: given a real vector space V with complex structure J: V to V, the resulting complex vector space (V,J) is complex-linearly isomorphic to the (+i) eigenspace V^(1,0) of J on the complexification V_C of V (namely V_C = V tensor_R C). We call this (+i) eigenspace the (1,0) part, or the holomorphic (with respect to J) part, of the complexification V_C. V_C splits into (1,0) part plus its complex conjugate V^(0,1), (0,1) part or equivalently the (-i) eigenspace for J. The manifold version: the complexified tangent bundle, and its corresponding holomorphic and anti-holomorphic parts (with respect to an almost complex structure). Auroux lecture 12, Cannas da Silva chapter 14, Wells Chapter I Sec. 3.