Math 474: Introduction to Differential Geometry





WHEN: MWF 2– 2:50 pm

WHERE: ENG E204 // in the oval with good weather

WHAT: An introduction to differential geometry of curves and surfaces in three dimensional space.

TEXTBOOK: Differential Geometry of Curves and Surfaces (Manfredo Perdigao do Carmo)








PDF file

LaTeX source file


Office hours : the official office hours for this class are right after class. However, if these times are not convenient for you, you are very welcome to make an appointment and come ask questions, make comments, or just chat. You can also try showing up at my door anytime. But I might tell you to come back at another time if I am immersed into something else.

The tables of the law for this class are contained in the following document:


Homework:  math is not a sport for bystanders. Getting your hands dirty  is important  to make sure that things sink in and you are not just spending a semester assisting to my creative rambling. Homework will be due pretty much every Friday (but see below for the up-to-date information). I will NOT grade all the problems I assign. If one of the problems you feel unsure about doesn’t get graded, please don’t go “Whew! Lucky one!” but rather come ask me about it. The point is you understanding, not pretending to!!

READ CAREFULLY!! In fact, here is the homework grading policy. For every assignment, you are supposed to write up all problems. However you need to identify two questions: the one that you feel have done best on (which will be graded), and the one where you feel you have done worst on (which will be looked at and commented but not graded).

1.      Due September 6th.  a. Find a parameterized differentiable curve whose trace is the nodal plane cubic (z=0, y^2=x^3+x^2). Hint: use as a parameter the slope of a tangent line through the origin.

b.      Exercises 2,3,4 page 5 and 2 page 7

2.      Due September 13th   Exercise 6 page 8,  Exercises 2,4,6,9,13 pages 22-25.

3.      Due September 27th   Exercises  1,4,8, 16 pages 65-67.

Prove the following theorem. If f: R^{n+2}àR^n is a differentiable function and a is a regular value, f^{-1}(a) is a regular surface in R^{n+2}. (aka you can show the existence of local parameterizations.)

4.      Due October 7th                  Exercises 3,10 pages 80,81. Note: read the paragraph on surfaces of revolution (pages 76-78) before doing exercise 10!

5.      Due October 11th       Problems 4,5,6,7,8 on the handout on Metrics

6.      Due October 18th       Problems 9,10,11,12,13,14,15 on the handout on Metrics

7.      Due Nov 1st                Homework on the second fundamental form Solutions

8.      Due Nov 8th               More fun homework

9.      Due Nov 22nd             Exercises 4 and 9, page 237. Note: this hwk is very short, but I imagine it will give you a bit to think. Once you find the solution – try to ponder it a bit, and think about what general mathematical strategy it is pointing to.

10.  Due Dec 2nd               Homework on Euler Characteristic




These are  NOT lecture notes! Or, well, I guess they are, but they are just what I scribble down for myself in order to plan a lecture and recall how things happened. So they are by no means complete, pretty, nor do I promise I will absolutely be able to keep them updated. But if they can be of use you are welcome to refer to them.

Lecture Plan 474

Here is the worksheet on Metrics (LaTeX file)

Some notes on  Gauss-Bonnet and the Index Theorem. Disclaimer: they are rough and meant to be accompanied by the lecturer’s explanation on notation and contextualization (there is some abuse of notation here and there which is often standard in the field but nonetheless makes me a bit uncomfortable to have in writing without pointing it out!). However there are plenty colourful pictures which I hope help.