# An Introduction to Differential Geometry

## Section Information

 Section Time Location Instructor Office Hours (Cupples I, Room 17) 1 MWF 3:00 PM - 4:00 PM Cupples I 218 Renato Feres TTh 1:00 PM - 3:00 PM

 Please include [Math495] in the subject line of any email message that pertains to this course. This will help avoid that I accidentally delete your still unread message. My e-mail address is feres@math.wustl.edu.

## Text

For the core topic of the course, which will be about the geometry and topology of surfaces, we will use Differential Forms and Applications, by M. do Carmo
Springer, 1994

We will supplement the textbook with course notes to be handed out (or made available online) throughout the course.
The notes will contain more than we will actually cover. I will tell in class the relevant sections.
Course notes for Math 407

## Topics to be covered

• Multilinear and tensor algebra;
• Review of calculus in Rn;
• Geometry of Frenet curvers;
• Calculus of tensor fields and differential forms;
• Manifolds;
• Integration of forms and Stokes' theorem
• The Euclidean group and other Lie groups
• The geometry of surfaces
• The Gauss-Bonnet theorem and topology of surfaces
• Elements of Riemannian geometry

We plan to have 9 weekly homework assignments, one midterm, and one final exam. They will all have the same weight. The exams will be take-home and will not be fundamentally different from the homework sets. The main difference is that I plan to grade the exams mostly myself (with perhaps some TA help) and will likely be more strict in grading them. The mid-term homework-exam will be due 2/20/2015 and the final homework-exam will be due 4/24/2015.

The final score will be the simple mean (with equal weights) of the top 10 scores (adjusted so that the maximal score is 100). This means that you can drop the least score of the 11.

The final score will be translated into a letter grade of A, B, C, D, F (with plus and minus shadings) in a way that is not harsher than the following table (if the grade distribution of the whole class is significantly lower than usual, then "curving" may be considered, but it is unlikely to be needed):

 Numerical Range Letter Grade [85, 100] A [70, 85) B [60, 70) C [50, 60) D [0, 50) F

In assigning pluses and minuses, I will divide the final scores in each letter roughly evenly. Please note: by this I do not mean, for example, that A- corresponds to the interval [85, 90) or that A corresponds to [90, 95). Rather, the scores in each letter range indicated in the table will be divided in roughly equal numbers so that the number of scores corresponding to A-, A, A+ will be approximately the same. Therefore, the exact grade cut-offs for the signs will not be known until all the scores are in.

## Homework

The plan is to have a total of 11 HW/exams assignments. Of these 11, the 10 best scores will count towards the total grade score.

Assignments should be turned in on the due date (Fridays) at the beginning of class.
You are allowed and encouraged to collaborate on your assignments.
Late assignments will not be accepted except under very special circumstances and never after the solutions have been posted.

## Course plan

 Week Comments Jan 12 - Jan 16 Material from notes Jan 21 - Jan 23 Homework 1: 1/23 Solutions Jan 26 - Jan 30 Homework 2: 1/30 Solutions Feb 02 - Feb 06 Homework 3: 2/6 Solutions Feb 9 - Feb 13 Feb 16 - Feb 20 Homework 4: 2/20 Solutions Feb 23 - Feb 27 Homework 5: 1/27 Solutions Mar 02 - Mar 06 Homework 6: 3/6 Solutions Mar 16 - Mar 20 Mar 23 - Mar 27 Homework 7: 3/27 Solutions Mar 30 - Apr 03 Homework 8: 4/3 Solutions Apr 06 - Apr 10 Homework 9: 4/10 Solutions Apr 13 - Apr 17 Apr 20 - Apr 24 Homework 10: 4/24 Solutions

Renato Feres
feres@math.wustl.edu
Cupples I, Room 17
(314) 935 - 6752 (office phone)

Department of Mathematics
Washington University
Campus Box 1146
Saint Louis
Missouri, 63130 USA.

Last Updated: January 10, 2015