Differential Geometry, MAT733

In the following syllabus, homework is written below the lesson when it is assigned and should be completed before the next meeting. Starred assignments form part of the projects. Students should come to office hours to go over homework questions.

This class meets 11-1:50 twice a week in the same room as the 3 credit undergraduate course MAT432. Students taking MAT432 will be dismissed from some lessons early and will skip other lessons altogether. Their exams will be easier and their homework easier. They will not be learning the proofs of all the theorems, nor will they learn the intrinsic definition of a manifold, or the concept of a differential form. Graduate Students may find the beginning of the course easy.

MAT 733 Syllabus: (homework will be posted regularly)

• Mon 1/28: Open and Closed sets in Euclidean Space (handout)
  Read and do exercises on the handout.
  Quantifiers Review Sheet
• Wed 1/30: Open and Closed Sets in Metric Spaces 1.2 (Spivak 1-10)
  Read pages 2-78 in 1.2 and the handout and this, Do 1.2.1 (1-3), 1.2.5, 1.2.15
  Email me about 1/28 handout radius for exercise 6
  Practice Project on Open Sets is due 2/4 (voluntary)
  Handout exercises 6,7, Bloch 1.2.1 (1), 1.2.1 (2)
• Mon 2/4: Unions and Intersections of sets 1.2
  read pages 7-11 of 1.2, read the proof of Lemma 1.2.3 closely
  Do 1.2.2, 1.2.4, 1.2.5, 1.2.7
• Wed 2/6: Continuous Maps 1.3 (Spivak 11-14) Homeomorphisms 1.4
  Read 1.3, Do 1.3.6, 1.3.8, 1.3.9, Read 21-22, Do 1.4.1, 1.4.2 Read homeomorphism handout
  Project 1 on Continuity and Open sets is due 2/11:
  1.2.1(1), 1.2.7, 1.3.6, 1.4.1 and
  1) Find a homeomorphism from the open unit disk to the open upper half sphere. You do not need to prove it is a homeomorphism but write an explicit formula for f(x,y) as a vector valued function and find f inverse and explain why is it a homeomorphism using the fact that you know some functions are continuous from calculus I and using lemma 1.3.8. There are infinitely many correct solutions to this problem!
• Mon 2/11: Homeomorphisms 1.4
  Read Prop 1.3.3 and its proof, read Lemma 1.4.3, Do Exercise 1.4.2, Read pages 22-26 on quotient maps and gluing, Do 1.4.6 (1)
• Wed 2/13: Compactness 1.6
  Read pages 34-38 and statement of Heine Borel Theorem, Review Cauchy sequences from undergraduate analysis, Do 1.6.1, 1.6.3 (hint just do it for any two metric spaces X and Y and move the open cover from one space to the other using Lemma 1.4.3), 1.6.6 (use 1.6.3), find five examples of compact sets and 5 examples of noncompact sets in the plane.
  Project 2 on Compactness is due 2/25:
  1.2.9, 1.2.15 (see page 4 for defn of closed ball), 1.6.1, 1.6.3, 1.6.6, try 1.6.5
• Wed 2/20: Mapping Compact and Connected Sets 1.5-1.6
  Read proofs of Thms and Props: 1.6.8 1.6.10, 1.6.11, 1.6.12, 1.6.13, 1.6.14
  Do 1.6.11, and work on the project.
• Mon 2/25: Contraction Mapping Principle
• Wed 2/27: Review (Spivak Chapter 1)
  This exam is on 1.2-1.6 of Bloch and Projects 1 and 2. You will need to know proving techniques including quantifiers, first lines of proofs by contradiction, how to prove something is an open set, how to prove one set is inside another, how to prove a map is continuous using balls and using epsilon deltas. There will not be full length proofs on the exam, just questions regarding first and last steps and such things, how to draw a diagram that will help you select a ball and so on. You should also be able to identify sets in a plane which are closed, bounded, open, compact, and homeomorphic without providing proofs. Sample Exam You do not need to know how to do a compactness proof or the contraction mapping material for the exam.
• Mon 3/3: Exam I
• Wed 3/5: The Space of Continuous Functions (handout)
  Do (b)-(j) on the handout. You may need to refer to your undergraduate analysis textbook for some theorems to fill in justifications.
• Mon 3/10: Smooth curves 4.3-4.4
  Read 4.3-4.4 (we will do 4.2 later), Do 4.3.1*, 4.3.6*, 4.3.7*, 4.3.8*, 4.3.9*, 4.4.1, 4.4.2*, 4.4.4
• Wed 3/12: Curvature and Torsion 4.5, Smooth Functions 4.2
  Read 4.5, Do 4.5.3, 4.5.4*, Read page 167-168. We will do the rest of 4.2 later.
  Project 3 on Curves is due 3/26 (starred problems assigned 3/10-3/12), bring questions on 3/17.
• Mon 3/17: Ordinary Differential Equations 4.2 ~
• Wed 3/19: Fundamental Theorem of Curves 4.6 ~
• Wed 3/26: Chain and Product Rules in Vector Calculus (Spivak 19-33))
  Go to google book search and bring up Marsden and Weinstein Calculus. Section 15.4 is Matrix Multiplication and the Chain Rule. Do problems 21, 25 and 31. This would be a good time to read Spivak since it is much clearer than Bloch on this material.
• Mon 3/31: Inverse Function Theorem 4.2 ~ (Spivak 34-39)
• Wed 4/2: Smooth Surfaces 5.1-5.3 and Manifolds (Spivak 109-114)
  Project 4 on Surfaces is due 4/14
  5.2.5, 5.2.3, 5.3.3 can be started now
  two out of 5.4.3, 5.4.4, 5.4.5, 5.5.7, 5.5.8 can be done later
• Mon 4/7: Implicit Function and Rank Theorem 4.2 (Spivak 40-44)
Read Spivak handout for implicit function theorem and focus on n=2, m=1, and application to ellipsoid, Then read defn of diffeom on page 168 of Bloch, Defn of surface smooth on page 210 of Bloch, Then Lemma 5.2.7, then Change of coords on p206, Then Prop 5.2.5 Then Rank Theorem 4.2.2, Then reread everything in the reverse direction.
• Wed 4/9: Tangent and Normal Vectors 5.4-5.5 (Spivak 86-88)
Do two out of 5.4.3, 5.4.4, 5.4.5, by Monday
• Mon 4/14: Charts 5.2 and Metrics 5.4
  Do 5.5.7, 5.5.8. Final resubmission of Projects 1-3 is due Wed 4/30.
• Wed 4/16: Review of Compactness and C([0,1])
• Spring Recess
• Mon 4/28: No meeting
• Wed 4/30: Length, Area and Fubini's Theorem 5.8 (Spivak 56-62) Integration and Measure Zero (Spivak 46-55, 63-66)
  Extra Credit Project on Area is due 5/12: Before trying the project read 5.5 and do 5.5.7 and 5.5.8. Then read 5.8 and the project is: 5.8.1,5.8.3,5.8.5, a and b where
  (a) Let f(r,s)=(rcos(s), r sin(s)) define a chart for the plane with domain (0,2pi)x(0,R). The image is most of a disk of radius R. Compute the metric and then find the area of the disk using Defn 5.8.1. Does the formula in the integral look familiar? Does rdrds remind you of something from vector calculus? What is the area of a sector in the plane?
  (b) Compute the area of a rotationally symmetric surface as described by the chart in 5.3.1.
• Mon 5/5: Fubini's Theorem 5.6 ~ Tensors and Forms (Spivak 79-84, 116-119)
  Extra Credit Project on Area is due 5/12. Final resubmission of project 4 is 5/12.
• Wed 5/7: Method of Lagrange Multipliers proven
  Read the description of the final given below and look over the Updated Sample Final. Notice that completing Project 4 and doing the Extra Credit Area Project are both good practice for this final and are due Monday.
• Mon 5/12: Review
  The final will be on the entire semester of the course and students may also wish to review key topics from linear algebra and vector calculus to succeed. In addition to topics covered on the first exam, students need to know 4.2-4.6, 5.1-5.5 and 5.8, Projects 3 and 4 and the Area Project. There will be a problem where one is asked to compute area. It is important to be able to find a monge patch for a given surface, to be able to compute a change of coordinates map given two coordinate charts, to be able to verify a given chart is a patch (checking the columns of Dx are independant using the cross product), to find the normal vector to a surface with a given patch, to find the area with a given patch. You may also be asked to find charts for other surfaces (like rotationally symmetric charts). You should be familiar with all the examples in the textbook. You may be asked to draw a graph of a curve or surface if you are given the formula for it. You must be able to compute the length of a curve and find the formula for a curve. There will be a proof on the exam for graduate students (extra credit for undergrads). The proof will be a proof of continuity or open sets and should be done using the definitions of these concepts.
• Wed May 14: Study Session, no meeting,
• Mon May 19: extra office hours and pick up Project 4
• Final Wed May 21