Differential Geometry, MAT733
|Differential Geometry, MAT733
Grading Policy: 4 Projects: 10% each... Midterm Exam: 20%...
Quizes are on Mondays
Prerequisites: Vector Calculus and Linear Algebra
Meetings: 11-1:50 Mon Wed
Bloch, A first course in geometric topology and differential geometry
Spivak, Calculus on Manifolds
|Professor C. Sormani
Office Hours: Monday Wednesday 10-11 am, 3-4 pm
Office: Gillet Hall 200B
Email: sormanic (at) member.ams.org
Course Description: Metric and Euclidean Geometry,
compactness, connectedness, contraction mapping,
torsion and curvature of curves, fundamental theorem of curves,
surfaces, inverse and implicit function theorems,
manifolds, differential forms and tensors, curvature,
area and geodesics.
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.
Grads and Undergrads Together:
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 2/4: Unions and Intersections of sets 1.2
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)
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)
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
Mon 2/11: Homeomorphisms 1.4
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!
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])
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