Instructor | Sheel Ganatra | TA | Tianle Liu |

Office | KAP 266D | Office | |

sheel (dot) ganatra (at) usc (dot) edu (this is the best way to reach me) | tianleli (at) usc (dot) edu |

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).

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 review of Calculus on R^n: inverse and implicit function theorems
- Topological and Differentiable manifolds and maps between them. Sard's theorem. Immersions, Submersions, and embeddings.
- The tangent bundle: vector fields, distributions, and Frobenius' theorem
- Calculus on manifolds with differential forms, tensors, and vector fields.
- Integration. Stokes' theorem and de Rham cohomology.
- (time permitting) Vector bundles and operations on them; the normal bundle of an embedding, and the tubular neighborhood theorem.
- Degree theory and (time permitting) transversality theory.
- (time permitting) An introduction to
*Riemannian geometry*: Riemannian metrics, connections, and curvature

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

- 50% Homework assignments,
- 20% midterm (takehome),
- 30% (in class) final exam.

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 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 |