Math 474:
Introduction to Differential Geometry
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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)
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LaTeX CHEATSHEET:
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:
Syllabus
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.
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.