# Differential Geometry

 Differential Geometry, MAT432 Grading Policy: 4 Projects: 10% each... Midterm Exam: 20%... Final: 40%... Quizes are on Mondays Prerequisites: Vector Calculus and Linear Algebra Text: Bloch, A first course in geometric topology and differential geometry Professor C. Sormani Office Hours: Monday Wednesday 10-11 am, 3-4 pm Office: Gillet Hall 200B Email: sormanic (at) member.ams.org Webpage: http://comet.lehman.cuny.edu/sormani

Course Description: Metric and Euclidean Geometry, torsion and curvature of curves, fundamental theorem of curves, surfaces, inverse and implicit function theorems, curvature, area and geodesics.

Homework:
In the following syllabus, homework is written below the lesson when it is assigned and should be completed before the next meeting. Starred problems may form parts of the projects. Students should come to 10 am office hours to go over homework questions.

This class meets 11-12:50 twice a week in the same room as the 4 credit graduate course MAT733. Since MAT432 is only 3 credits, not as much work will be expected of you. Certain lessons will be for graduate students only (marked as "no meeting" below) and undergraduates will be dismissed early from many lessons (marked with a ~ below). The graduate students will be learning the proofs of all the theorems and will have more difficult exams and homework.

MAT 432 Syllabus: (homework will be posted regularly)

• Mon 1/28: Open and Closed sets in Euclidean Space (handout)
Read handout pages 1-3 and do exercises 1 and 2.
Then read handout pages 6-8 and do exercises 4,5,6,7.
Do the quantifiers review sheet before next Monday.
The handout is an excerpt of Marsden Elementary Classical Analysis which also explains open and closed sets.
• Wed 1/30: Open and Closed Sets in Metric Spaces 1.2
read pages 2-7 in 1.2, handout and this , do 1.2.1 (1), 1.2.1 (2),
Project 1 on Open Set Proofs is due 2/4:
Exercises 6,7 of handout, 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 1.2.4, 1.2.8
• Wed 2/6: Continuous Maps 1.3
Read 13-17, focus on Example 1.3.4,
use an epsilon delta proof to show f(x)=3x+7 is continuous at x=2 (be sure to give a precise formula for delta),
Very Difficult: g(x)=cos(x) is continuous at any x0 (warning the formula for delta will depend on x0 and on epsilon),
Read 18-19, Do 1.3.6 (hint use lemma 1.2.9),
Project 2 on Continuity Proofs is due 2/20
1) Let f:[1,4] --> [2, 17] be defined as f(x)= 5x-3
a) Show f is continuous on [1,4] using epsilon deltas
b) Find the inverse of f
c) Show the inverse is continuous on [2,17] using epsilon deltas
d) Show f is a homeomorphism
2) Find a homeomorphism from the closed upper half sphere to the closed unit disk. 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)
For fun see geometry games.
• Wed 2/13: Closed and Compact Sets 1.6
Read pages 9-11, Do 1.2.8, Read pages 34-5 (this is difficult, go on even if it is confusing and come back to it), Read defn of bounded on page 37, Read page 38 and the statement of Heine-Borel Thm, Give 5 examples of compact sets in the plane and 5 examples of sets which are not compact. Do not worry about proofs.
• Wed 2/20: Mapping Compact Sets 1.6
Read Theorems 1.6.10, Prop 1.6.11, Prop 1.6.12, Thm 1.6.13 and know these statements, Pick up Project 2 from my door and be resubmit it on Monday 2/25 so it can be graded again by 2/27.
• Mon 2/25: Study Session No Meeting,
• Wed 2/27: Review
This exam is on 1.2-1.6 of the book 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
• Mon 3/3: Exam I
• Wed 3/5: No meeting
• 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)
• 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
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.
• Mon 3/31: Inverse Function Theorem 4.2 ~
• Wed 4/2: Smooth Surfaces 5.1-5.3 ~
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 ~
• 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
• do two out of 5.4.3, 5.4.4, 5.4.5, this is part of the project due Monday.
• Mon 4/14: Charts 5.2 and Metrics 5.5
Over break try 5.5.7 and 5.5.8 possibly to add into Project 4. Final resubmission of Projects 1-3 is Wed 4/30.
• Wed 4/16: no meeting
• Spring Recess
• Mon 4/28: no meeting
• Wed 4/30: Length, Area and Fubini's Theorem 5.8
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 proven 5.8
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 this 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.