math214:start

# Math 214: Differentiable manifolds

UC Berkeley, Spring 2020

Lecture: MWF 11-12, at 9 Lewis Hall

Zoom Meeting: https://berkeley.zoom.us/j/515298392
Zoom Office hour: Enter Office Wednesday 2-4pm, Thursday 1-2pm, at PMI number: 881-910-2324. Please also join the zoom channel, “Math 214”. If I am not in 'office', you can leave a message there.

## Instructor

Peng Zhou: pzhou.math@berkeley.edu
Office: 931 Evans Hall
Office Hour: M 12-1, W 2-4pm. MWF: 10-11am.

GSI: Alex Sherman.
Office Hour: Tue 2-4pm at 1057 Evans Hall

## Syllabus

This is a first year graduate differential geometry course. Our course roughly has three parts:

• Part I: “vocabulary and grammar”. We learn the basic definition, constructions and theorems. Things like: the definition of smooth manifold, vector fields, differential forms, Lie group and Lie algebra, principal bundles.
• Part II: Riemanian Manifold. In this part you will encounter metric, connection, curvatures. We will see the connection to general relativity.
• Part III: Vector bundles, Connections and Characteristic Classes. Chern-Weil theory.

Piazza This is an online Q&A platform, you can even post question anonymously.

The total grade will be 60% homework and 40% take home final. For homework, you are encouraged to work in group, and discuss as much as possible, but you should write your own solutions.

1. Two loweste scores of homework will be dropped.
2. HW is due by 12noon Friday, in class or by my door folder.
3. If you need to turn in late, you need to have my permission first. If granted, you can turn in to the GSI's office during the Tuesday office hour.
4. No electronic submission. You need to submit a hard copy or printed copy. It is inconvenient to grade HW electronically.

## References

Our official textbook is John Lee's Introduction to smooth manifolds, 2nd edition. It has all the details spelled out. However, Lee's book does not cover characteristic classes. I will follow other textbook or notes, for example Nicolascu's online note Chapter 8.

• Warner. Foundations of Differentiable Manifolds and Lie Groups. If you want a concise introduction, try this one.
• Milnor, Topology from the differentiable viewpoint. Milnor is exemplary in clear and concise math writing. The section 1-4 are relevant for our class. Also his Morse Theory contains part relevant for our Riemannian geometry part.
• Bott and Tu, Differential forms in algebraic topology. It takes hands-on approach to algebraic topology (over $\R$) using de Rham differential forms. It has special appeal to physicists.
• Gallot-Hulin-Lafontaine, Riemannian Geometry 3rd ed. Despite the title, the book starts from the basic differential manifold. The first chapter roughly corresponds to our Part I. And our Part II will be a small subset there.
• Kobayashi-Nomizu. Foundations of Differential Geometry Vol 1. The definite reference.

• Frank Morgan, Riemannian Geometry: A Beginners Guide. The short book is a fun reading, with many pictures and illustrates many important ideas.
• Marcel Berger, A Panoramic View of Riemannian Geometry. (2003) An encyclopedia written by one of the top expert, leading you to the frontier quickly.

We borrow heavily from Prof. Hutchings' course website, where you can also see comments on other references.

## Latex

It is encouraged that you use Latex to submit the homework. An easy way to type latex without having to install the software is to use overleaf, an online tex editor-compiler.

Here is a sample latex template. Try tinkering with it to suit your need.

## Lectures

• 2020-01-22, Wednesday: definition and examples of smooth manifolds.
• 2020-01-24, Friday: smooth structures and smooth function. Paracompactness and Partition of unity.
• HW1: Ex in Lee's book: 1-4, 1-6, 2-1, 2-9, 2-14. Due 1/31(Friday) in class or drop to my office.
• 2020-01-27, Monday: tangent space and differential of smooth maps
• 2020-01-29, Wednesday: The tangent bundle. Immersions, embeddings, and submersions.
• 2020-01-31, Friday: Sard's theorem
• HW2: Ex 3-6, 3-7, 3-8, 4-4, 4-6 (Due 2/7)
• 02-03: finish Sard theorem
• 2020-02-05 Wednesday: Submanifold and Whitney Embedding Theorem.
• 2020-02-07, Friday: Vector Field, Commutator, Integral Curve, Flow.
• HW3: Ex 5-1, 5-6, 5-15, 5-18. And read the proof of Whitney Embedding theorem for non-compact manifold, then sketch the outline of the proof.
• 2020-02-10, Monday: Whitney Approximation Theorem.
• 2020-02-12, Wednesday: Transversality.
• 02-14: Finish up Transversality
• HW4: 6-3, 6-5, 6-9, 6-11, 6-16(a,b,c,f)
• 2020-02-19, Wednesday: Vector Bundle and Examples Cotangent Bundle. Poincare Lemma for 1-form.
• 2020-02-21, Friday: Tensor power, Exterior Power.
• HW5: 9-19, 9-22, 10-7, 10-15, 10-18, 11-5, 11-7(a)(b), 11-11
• 02-24: Differential $k$-forms. Exterior derivative $d$.
• 02-26: Properties of $d$.
• 2020-02-28, Friday: Lie derivatives. Interior multiplication. Cartan formula.
• HW6: Read the proof of Thm 9.38 (p229), and prove Corollary 9.39. Ex 9-8 (hint: use Thm 9.20), 14-5, 14-6,
• 03-02: Finishes Cartan formula. Finishes discussion on exterior derivative.
• 2020-03-04, Wednesday: Lie Group. Ch 7 of Lee.
• 03-06: Discussion of HW. Constant Rank Thm of equivariant map. Lie algebra begin
• HW7:
1. Read Example 14.27 (p367) and do Ex 14.28
2. Lie Group: 7-2, 4, 9, 11
• 03-09
• 03-11: lecture note. Open subgroup. Generating set. Maure-Cartan form.
• 03-13: lecture note. Principal G-bundle. Homogeneous space. Associated Bundles. Basic definition.
• HW8:
1. Ch 7: 13
2. Ch 8: 19,22,28,31
• 03-16: Ch 21 quotient manifold. note
• 03-18: Continue with Ch 21, Fill in the details of proofs of quotient map theorems. note from lecture
• typo in the note: one page 2, right column. I wrote incorrectly that, “if $p \in K \cap gK$ then $p, gp \in K$”. The correct statement is that “if $g \in G_K$, then there exist $p \in K$ such that $g \cdot p \in K$”.
• 03-20: Ch 19, Distribution. covered up to Frobenius Thm (statement and sketch of proof)
• HW9: Ch 21: 1,5,9,16. Ch 19: 1. HW 9
• Week 10. From Wednesday on, we will start to use the lecture note of Nicolascu for topics about connections, denoted as [Ni]
• 03-30: Riemannian metric: an introduction with examples and no proofs.[Lee]
• Errata: During lecture, I answered a question “is the Haar measure of a Lie group the measure induced by the left-invariant metric?” The answer should be: true for a compact Lie group. See wiki for an example about the difference of left- and right-invariant Haar measure when the group is non-compact.
• 2020-04-01, Wednesday: Connections on Vector bundle. Cartan's Moving Frame. [Ni] Section 3.3.1
• 2020-04-03, Friday: Parallel Transport, Curvature. [Ni] 3.3.2, 3.3.3
• HW 10 :
• Week 11:
• 2020-04-06, Monday: ipad noteHolonomy, interpretation of Curvature[Ni] 3.3.4
• Errata of ipad note: page 1, right column. It should be: if $A = \omega \otimes (e_\alpha \ot \delta^\beta)$, for $\omega \in \Omega^1(U)$ and $e_\alpha$ the local frame of $E$ previously chosen, and $\delta^\beta$ the dual frame of $E^*$, then $dA = d(\omega) \otimes (e_\alpha \ot \delta^\beta)$. This is because $d e_\alpha =0, d \delta^\beta=0$ by the definition of local connection $d$ on $E|_U$.
• 04-08: ipad note Connection on Tangent Space. Levi-Cevita connection.
• 04-10: ipad note. Levi-Cevita connection for bi-invariant metric on Lie group. Begin Geodesic equation.
• Homework 11 Due Sunday 8pm. Hint
• Week 12
• Week 13:
• Calculus of Variation. Jacobi Field. Reference: Milnor Morse Theory (available online). Lee, Riemannian manifold, Ch 10 (available through library online, or here), Nicolescu's lecture note (following Milnor).
• 04-20: ipad note (typo about the critical point for $f(x,y,z) = xyz$, it should be $\{yz=0, xy=0, zx=0\}$ which is a union of three coordinate axixes.
• 04-22: ipad note. This follows [Ni]'s lecture note, section 5.2.