Lecture Notes

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======  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: [[https://berkeley.zoom.us/my/pzhoumath | 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. 

Please sign up at https://piazza.com/berkeley/spring2020/math214

** Grading **

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.

About Homework
   - Two loweste scores of homework will be dropped. 
   - HW is due by 12noon Friday, in class or by my door folder. 
   - 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.
   - 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 [[https://www3.nd.edu/~lnicolae/Lectures.pdf|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. 

The following are not textbooks, but for additional reading
  * 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' [[https://math.berkeley.edu/~hutching/teach/214-2015/index.html|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 [[https://overleaf.com | overleaf]], an online tex editor-compiler.

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

===== Lectures =====

  * [[01-22]]: definition and examples of smooth manifolds. 
  * [[01-24]]: 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.
  * [[01-27]]: tangent space and differential of smooth maps
  * [[01-29]]: The tangent bundle. Immersions, embeddings, and submersions. 
  * [[01-31]]: Sard's theorem
  * HW2: Ex 3-6, 3-7, 3-8, 4-4, 4-6 (Due 2/7)
  * 02-03: finish Sard theorem
  * [[02-05]]: Submanifold and Whitney Embedding Theorem. 
  * [[02-07]]: 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. 
  * [[02-10]]: Whitney Approximation Theorem. 
  * [[02-12]]: Transversality.
  * 02-14: Finish up Transversality
  * HW4: 6-3, 6-5, 6-9, 6-11, 6-16(a,b,c,f)
  * [[02-19]]: Vector Bundle and Examples Cotangent Bundle. Poincare Lemma for 1-form. 
  * [[02-21]]: 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 $d$.  
  * [[02-28]]: 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. 
  * [[03-04]]: Lie Group. Ch 7 of Lee. 
  * 03-06: Discussion of HW. Constant Rank Thm of equivariant map. Lie algebra begin
  * HW7: 
     - Read Example 14.27 (p367) and do Ex 14.28
     - Lie Group: 7-2, 4, 9, 11
  * 03-09
  * 03-11: {{ :math214:note_11_mar_2020.pdf | lecture note}}. Open subgroup. Generating set. Maure-Cartan form.  
  * 03-13: {{ :math214:note_13_mar_2020.pdf | lecture note}}. Principal G-bundle. Homogeneous space. Associated Bundles. Basic definition. 
  * HW8: 
    - Ch 7: 13
    - Ch 8: 19,22,28,31
  * 03-16: Ch 21 quotient manifold. {{ :math214:note_16_mar_2020.pdf | note}}
  * 03-18: Continue with Ch 21, Fill in the details of proofs of quotient map theorems.  {{ :math214:note_18_mar_2020.pdf | 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. [[hw9-sol]]
  * Week 10. From Wednesday on, we will start to use the [[https://www3.nd.edu/~lnicolae/Lectures.pdf| 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[[https://en.wikipedia.org/wiki/Haar_measure#The_modular_function | wiki ]] for an example about the difference of left- and right-invariant Haar measure when the group is non-compact. 
    * [[04-01]]: Connections on Vector bundle. Cartan's Moving Frame. [Ni] Section 3.3.1
    * [[04-03]]: Parallel Transport, Curvature. [Ni] 3.3.2, 3.3.3
    * HW 10 : 
      * [Lee]: 13.16, 13.19, 13.20. and [[hw10 | two problems on connection and curvatures]] 
  * Week 11: 
    * [[04-06]]: {{ :math214:note_6_apr_2020.pdf |ipad note}}Holonomy, 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: {{ :math214:note_8_apr_2020.pdf |ipad note}} Connection on Tangent Space. Levi-Cevita connection. 
    * 04-10: {{ :math214:note_10_apr_2020_3_.pdf | ipad note}}. Levi-Cevita connection for bi-invariant metric on Lie group. Begin Geodesic equation.
    * [[hw11 | Homework 11]] Due Sunday 8pm. [[hw11-hint]]
  * Week 12
    * 04-13: {{ :math214:note_13_apr_2020_2_.pdf |ipad note}}
    * 04-15: {{ :math214:note_15_apr_2020_3_.pdf |ipad note}}
    * 04-17: {{ :math214:note_17_apr_2020_2_.pdf |ipad note}}
    * [[hw12| Homework 12]] Due Sunday 8pm. 
  * Week 13: 
    * Calculus of Variation. Jacobi Field. Reference: Milnor Morse Theory (available online). Lee, Riemannian manifold, Ch 10 (available through library online, or {{ :math214:lee_ch_10_rm_mfd.pdf |here}}), Nicolescu's lecture note (following Milnor). 
    * 04-20: {{ :math214:note_20_apr_2020_2_.pdf |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: {{ :math214:note_22_apr_2020_2_.pdf |ipad note}}. This follows [Ni]'s lecture note, section 5.2. 
    * 04-24:  {{ :math214:note_24_apr_2020_3_.pdf |ipad note}}
    * [[hw13]]
  * Week 14: de Rham cohomology
    * 04-27: {{ :math214:note_27_apr_2020_2_.pdf | ipad note}}
    * 04-29: {{ :math214:note_29_apr_2020_2_.pdf | note}} Poincare duality. {{ :math214:mv-sequence.pdf |Excerpt from Bott-Tu}}, p23-24, on MV sequence. 
    * 05-01: {{ :math214:note_1_may_2020_2_.pdf | note}}  Singular, Cech, Morse cohomology. 
  
 ** [[hwsol | Students Homework Solutions]] **


===== Final =====
[[final]] and [[final-solution]]
Lecture Notes

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