## Math 140 - Metric differential geometry - Spring 2004

Solutions to homework (by Maksim Maydanskiy)

Instructor: Bjorn Poonen

Lectures: MWF 8-9am, 9 Evans Hall

Course Control Number: 54930

Office: 879 Evans, e-mail: poonen@math

Office Hours: (starting 1/16/04) MWF 9:30-10:30am, TuTh 1:30-2:30pm, or by appointment (tentative)

Grader: Maksim Maydanskiy, mailbox on the 9th floor of Evans Hall near the north elevators.

Prerequisites: Math 104 and Math 110, or permission of instructor. (Ignore what the general catalog says.) Students having had Math 54 but not Math 110 can still take the course, but should be prepared to spend a little extra time learning some topics such as self-adjoint linear maps and quadratic forms.

Syllabus: This course uses multivariable calculus to study the geometry of curves and surfaces, with the emphasis on the latter. We will cover most of Chapters 1-4 in the do Carmo text. Some of the highlights will be Gauss's "Theorem Egregium", which states that the Gaussian curvature of a surface is intrinsic (independent of the way the surface is embedded), and the Gauss-Bonnet theorem, which relates area on a surface to its curvature. In order to focus on geometric ideas, we will not develop the technical machinery of differential forms in full generality.

Required Text: do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, 1976. Errata

Recommended Reading: Struik, Lectures on classical differential geometry, second edition, Dover, 1988. (a little old-fashioned, but good and inexpensive)

Exams: There will be midterm exams in class on Monday, February 23 and Monday, April 5. The final exam will be Friday, May 14, 8-11am in 60 Evans Hall.
Midterm 1 (solutions)
Midterm 2 (solutions)
Final (solutions)

Grading: 35% homework, 15% first midterm, 15% second midterm, 35% final. Each homework grade below the weighted average of your final and midterm grades will be boosted up to that average. The course grade will be curved. Click here for an example.

Homework: There will be weekly assignments due at the beginning of class each Monday. Late homework will not be accepted, but see the grading policy above. You should not expect to be able to solve every single problem on your own; instead you are encouraged to discuss questions with each other or to come to office hours for help. If you meet with a study group, please think about the problems in advance and try to do as many as you can on your own before meeting. After discussion with others, write-ups must be done separately. (In practice, this means that you should not be looking at other students' solutions as you write your own.) Write in complete sentences whenever reasonable. Staple loose sheets!