Math 401 - Differential Geometry - Fall 2005

********* Remarks for the Final Exam: **********
On #5, you can replace "diffeomorphic to a 2-dimensional disk" with "homeomorphic to a 2-dimensional closed disk {(x,y)| x^2 + y^2 <= 1}."
On #6, the region P is a compact region that is homeomorphic to a closed annulus {(x,y,z)| x^2+y^2=1,z in [0,1]}.

Catenoid		Costa-Hoffman-Meeks Surface

Comments and updates: · 8/26 - Homework 1 posted · 8/29 - List of links to Maple and Mathematica functions for Differential Geometry added below · 8/29 - Email me to get the code and address to buy Maple V 10 at the discounted price of $75.00 · 9/1 - Late homeworks will get no more than 50% credit and will not be graded · 9/14 - Added Maple files for Frenet-Serret Apparatus (see below) · 9/15 - Homework Solution 1 posted (more coming soon) · 9/27 - Due to Hurricane Rita, classes were cancelled September 22-27. The homework that was due on Friday, September 23rd will now be due on Wednesday, September 28th. If you need an extension until Friday, September 30th please email me. I will post a new homework on Wednesday, September 28th. This homework will be due a week from Friday. This homework assignment will be a little longer than normal. · 9/27 - Updated corrections to O'Neills book: Errata_Oneill_Diff_Geo_2nd.pdf · 9/28 - Homework Solutions 2 posted · 10/02 - Homework Solutions 3 posted · 10/2 - Red Sox win the AL Wildcard!!! · 10/11 - Solutions to Homework 4 and 5 posted · 10/11 - I will give out the midterm on Wed, 10/12 in class. It will be due in class on Wed, 10/19. The midterm will cover Sections 1.1-1.7, 2.1-2.7, and 4.1-4.3. If you miss class, please come get a copy in my office (you can call x3659 or email me to see if I am here at the time. If you cannot find me, email me to arrange a time for you to get the midterm. · 10/26 - Midterm solutions posted below. · 11/4 - Links to Galleries of Surfaces: Surfaces that are geometrically interesting and Generalized Monkey Saddle - from Weiqing Gu's page at Harvey Mudd College - 3D-XplorMath Surface Gallery from Richard Palais's website at UCI. · 11/18 - There will be one last homework given out on Monday. It will be due the last day of class! · 11/21 - A history of the celebrated Gauss-Bonnet Theorem: All the Way With Gauss-Bonnet and the Sociology of Mathematics (by D.H. Gottlieb) · 11/29 - All homework solutions posted except the one due on Friday! · 12/01 - Supplement to Friday's Movie "Outside In" including Movie clips and a brief history of sphere eversions (thanks to Geometry Center). · 12/01 - The Optiverse - another way to invert the sphere (6 min video). · 12/04 - All solutions posted.

Instructor:	Shelly Harvey
Email:	shelly@math.rice.edu
Office:	Herman Brown 424
Phone:	x3659
Office Hours:	M 1-2pm, R 2-3pm or by appointment

Course Information

Time and Place:	MWF 3-3:50 in Herman Brown 227
Required Text:	Elementary Differential Geometry by Barrett O'Neill (2nd edition), Chapters 1-2, 4-7
Recommended Software:	Maple V note: the appendix of O'Neill's book contains Maple and Mathematica computer formulas.
Prerequisites:	Math 212 (Multivariable Calculus) & Math 355 (Linear Algebra) or equivalent classes
Grader:	Soomin Kim (soomin@math.rice.edu)
Course Web Page:	http://math.rice.edu/~shelly/math401fall05/

In this course, we will study the geometry of curves and surfaces in three dimensional space using techniques from calculus. One way that we do this is to study geometric quantities associated to a curve or surface. You may already be aware of such a geometric quantity, namely the curvature (the length of the acceleration vector) at a point of a unit speed curve in R³. We will easily be able to show that a (nonconstant) curve with curvature zero at every point must be a line. This is a simple example of how the local behavior (the curvature at a point) determines the global behavior of the geometric object (the curve must be a straight line). We will study many other such results in this class. Perhaps the most remarkable results that we will encounter in this course, are the Poincare-Hopf Theorem (which relates the indices of a vector field at certain points of a closed surface to the number of handles of the surface) and the Gauss-Bonnet Theorem (which relates the total curvature of a closed surface to the number of handles).

I plan to cover Chapters 1-2 and 3-7 of O'Neill in this course. You should already be familiar with the material in Sections 1.1-1.4, 1.7, and 2.1-2.2. We will review this these sections during the first two days of of class and I expect you to read over these sections as soon as possible. I will post reading assignments on this web page. Unless otherwise stated, homeworks will be assigned every Friday (I will post them on the website) and will be due the following Friday. Late homeworks will be given no more than 50% credit and will not be graded. There will be one midterm and one final exam.

Links to Maple and Mathematica functions for Differential Geometry
Home page of Barrett O'Neill (Maple and Mathematica)
Maple V Application Center - Differential Geometry (Maple, a lot of packages/examples)

Maple Files (right click to save):
(1) Frenet Serret Animation from class (worksheet)
(2) You will need to download this (contains functions for curvature, torsion, normal, tangent and binormal) to use the above worksheet: frenet.m

Grading Policy:  Homework=50%, Midterm=25%, Final=25%.

Any student with a disability requiring accomodations in this course is encouraged to contact me after class or during office hours. Additionally, students also need to contact Disability Support Services in the Ley Student Center.

Reading Assignments: I plan to cover the following section(s) of O'Neill on the dates listed below.

Monday	Wednesday	Friday
August 22   § 1.1-1.3 (mostly review)	August 24   § 1.4,1.7 (some review)	August 26      § 1.7 (some review)
August 29   § 1.5	August 31   § 1.6	September 2   § 1.6, 2.1
September 5   no classes	September 7   § 2.2	September 9   § 2.2-2.3
September 12   § 2.3	September 14   § 2.3, 2.4	September 16   § 2.4
September 19   § 2.5	September 21   § 2.6, 2.7	September 23   no classes (hurricane)
September 26   no classes (hurricane)	September 28   § 4.1, 4.2	September 30   § 4.1, 4.2
October 3   § 4.3	October 5   § 4.3	October 7   § 4.4
October 10   no classes	October 12   § 4.4, 4.5	October 14   § 4.5
October 17   § 4.6	October 19   § 4.6	October 21   § 4.7
October 24   § 4.7	October 26   § 4.7	October 28   § 5.1
October 31   § 5.2	November 2   § 5.3	November 4   § 5.4, 5.5
November 7   § 5.6	November 9   § 6.1	November 11   § 6.2
November 14   § 6.3	November 16   § 6.3	November 18   § 6.4,6.5
November 21   History of Gauss-Bonnet, § 7.6 (no proofs)	November 23   § 7.7 (Thm 7.5, Cor 7.6, no proofs), 6.7	November 25   no classes (Thanksgiving)
November 28   § 7.7 (applications of GB), 6.8	November 30   § 6.8	December 2   (last day of classes)

Homework Assignments (pdf):

• Homework 1 (due September 2) - Solutions
  
• Homework 2 (due September 9) - Solutions
  
• Homework 3 (due September 16) - Solutions
  
• Homework 4 (due September 23) - Solutions
  
• Homework 5 (due October 7) - Solutions
  
• Homework 6 (due October 21) - Solutions
  
• Homework 7 (due October 28) - Solutions
  
• Homework 8 (due November 4) - Solutions
  
• Homework 9 (due November 11) - Solutions
  
• Homework 10 (due November 18, extension until Monday November 21) - Solutions
  
• Homework 11 (due on the last day of class, December 2) - Solutions