Description:

Some of the topics we hope to cover include:

- Parameterized and regular curves, arc length
- The Tangent-Normal-Binormal frame
- Fundamental Theorem of Space Curves

- Isoperimetric Theorem, Turning Tangents Theorem
- Cauchy-Crofton formula

- Mappings, the inverse function theorem, parameterizations
- Parameterized and regular surfaces, level surfaces
- Tangent Plane, differential, first fundamental form

- Orientation and area for surfaces in 3 space
- Differential forms and mappings of surfaces
- The Gauss map, Second Fundamental form, mean curvature
- Ruled and minimal surfaces
- Isometries, conformal maps
- Intrinsic versus extrinsic geometry, Gauss's
*theorema egregrium* - Geodesics, geodesic curvature
- Gauss-Bonnet theorem
- Exponential map
- Constant curvature surfaces

**Meets:** TTh 10:50AM- in Herman
Brown 423

**Instructor:** Robert Hardt, Herman Brown 430;
Office hours: 11-12 MWF (and others by appt.),

Email: hardt@rice.edu, Telephone: ext 3280

**Homepage:**
http://math.rice.edu/~hardt/401F03/

**Text: ** M.P. doCarmo, *Differential
Geometry of Curves and Surfaces,* Prentice-Hall, 1976.

**Handouts:**

Handout 2.

Handout 3.

Handout 4.

**Midterm Exam:** After class on
Thursday Oct 23, envelopes containing a takehome exam 1 will be
distributed. The completed exams should be returned at the beginning of
class the following Thursday, Oct. 30. (Envelopes containing the exam
are also outside of HB430). You should work on the exam for
2
(continuous) hours. You
may use books, notes, etc., but, of course, work alone. It will
cover
the class lectures through Oct 16, Do Carmo Page 143, and Handouts 1,
2, 3. In Handout 4, understand the statements (but not
necessarily the proofs) of the Inverse Function Theorem (especially
n=2) and Implicit Function Theorem (especially m=2 or 3, n=1)

**Final Exam:** After class on Thursday, Dec.4,
envelopes containing the takehome final exam will be distributed. The
completed exams should be returned to my office HB430 or the Math.
office HB220 on or before Wed. Dec.17. You should work on the
exam for at most 4(continuous) hours. You may use books, notes, etc.,
but, of course, work alone. It will cover the class
lectures, the handouts, and the DoCarmo sections which we studied
sometime in the course (see the list below) .

Chapter 2 : 2.1, 2.2, 2.3, 2.4, 2.5, 2.8, Appendix

Chapter 3 : 3.1, 3.2, 3.3, 3.4, 3.5B(Prop.1 only), Appendix

Chapter 4 : 4.1, 4.2, 4.3 (except p.236), 4.4(except Prop.4), 4.5

A. Gray, *Modern Differential Geometry of Curves and Surfaceswith
Mathematica, *CRC, 1997.

H. Guggenheimer, *Differential Geometry*, Dover,
1997.

D. Henderson, *Differential Geometry: A Geometric Introduction, *Prentice-Hall,1997.

R. Millman and G. Parker, *Elements of Differential Geometry*
,Prentice-Hall, 1997.

B. O'Neill, *Elementary Differential Geometry, 2nd Ed.*,
AcademicPress, New York, 1997.

J. Oprea, *Differential Geometry and its Applications, *Prentice-Hall,
1996.

D. Struik, *Lectures on Classical Differential Geometry, *Dover,
1988.

**Homework:** There will be a
homework assignment approximately every week. Here is the latest
homework assignment.
You are encouraged to discuss and work together on the homework
problems,
but each student is responsible for the final preparation of his or her
own solutions. At most one late homework assignment will be accepted,as
long
as it is turned in within seven days of its original due date. Homework
will
count for 40% of the final grade.

*Homework 1, due. Tues.,Sept.2* -- Sec.1.2: 1, 2,
Sec.1.3: 4,6,
Sec.1.4: 7,13

*Homework 2, due. Tues.,Sept.9* -- Sec.1.5: 1(a)(b)(c)(d)(e),
3(a)(b), 8(a)(b), 12(a)(b)

*Homework 3, due. Tues.,Sept.16* -- Exercise in Handout 1 (proving
a space of bounded continuous functions is a complete metric space),
Exercise in Handout 2 (integral equation form of an O.D.E.), Sec.1.5:
14 , Sec.1.7: 1, 2.

*Homework 4, due. Tues.,Sept.30* --Sec. 1.7: 1continued-Is there
one enclosing an area of 1 sqr. foot? Why or why not?, (**solution**) 3, 6 (**solution**),
Sec.2.2: 1, 3
(Hint: use Proposition3), 7(a),(b)(c).

*Homework 5, due. Tues.,Oct.7* -- Sec.2.3: 4, 10, Sec.2.4: 4,
13(a)(b), 15

(Hint: First consider how to do this problem in 1 lower dimension. Then
we would have a plane curve A(t) all of whose normal lines pass through
one point, which we may, for notational convenience, assume is (0,0)
But then the dot product A(t)*A'(t) would be identically 0
because A'(t) is tangent and (by assumption) A(t) is normal. .
This implies that |A(t)|^2 = A(t)*A(t)
would be constant so that A(t) lies on a circle. )

*Homework 6, due. Thurs.,Oct.16* -- Sec.2.5: 1(b), 2, 5, 9,
11(a)(b),
14(a)(b) (Hint: Find A and B so that grad f = A
x_u
+ B x_v by using the chain rule which gives the formulas
f_u
= df_p(x_u) = (grad f)* x_u and f_v = (grad f)* x_v.)
Sec.2.6:
7

*Homework 7, due. Tues.,Oct.23* -- Sec.2.4: 10, 11, Sec.2.5: 12,
Sec.3.2: 4, 8(a)(b)(c)

*Homework 8, due. Tues.,Nov* 11-- Sec.3.2: 5, 14, 17,
Sec.3.3: 2, 4, 21

*Homework 9, due. Tues.,Nov* 18-- Sec.3.4: 5, 8, 10ab,
Sec.3.5: 11(a)(b)(c)

(Hint for (b). Part(a) shows the parallel surface has the same normal.
One can use equations p.154(1) for the parallel surface to find
the corresponding a11, a21, a12, a22 . A shorter proof involves showing
that the corresponding principal curvatures of the parallel surface are
k1/(1-ak1), k2/(1-ak2) .)

*Homework 10, due. Tues.,Nov* 25-- Sec.4.2: 1, 14, 15,
Sec.4.3: 1, 4, 9

*Homework 11, due. Tues.,Dec* 2*-- Sec.4.4: 1(a)(Hint:
Consider a(s) wedge N(a(s)) where a(s) is a unit
speed parameterization of C and N(p) is the
normal to the surface at p .)`, 9, 15(a)(b)

*Homework 12, due. Tues.,Dec* 4*-- Sec.4.5: 1, 2(See p.157), 3
(Assume S is convex), 6(a)(c)

Late homework is accepted, but try not to get too far behind.

Thanksgiving Weekend:
I will be in California from Thursday through Monday night. Aaron
Trout (atrout@rice.edu) will be available in his office HB432 Monday
from 2-4 PM. From Tuesday on, I will be here. If you don't
catch me in the office, please e-mail me or phone (713-348-3280,
713-667-7946) to arrange a meeting.

See the webpage at http://math.rice.edu/~hardt/402S04/ and note that 402 is scheduled for TThurs.9:25-10:40 .

**Substitute:** Prof. Hardt will be absent for the lectures of
Sept.11,16,18. They will be given by Prof. Mike Wolf (HB404,
X6293, mwolf@rice.edu).

Extra Office Hours: I
will be in NYC in between the Oct.16 and Oct.21 classes. The grader
Aaron Trout (atrout@rice.edu) will be available Thurs.Oct.16, 2-3PM,
Fri.Oct.17,
2-3PM, Mon.Oct.20,2-3PM in HB430 or HB432.

Sunday, Oct.26. 7-9PM in HB430 for review for the midterm exam.

**Exam: ** There will be one midterm
exams worth 25% of the final grade. The final exam will be take-home,
and worth 35% of the final grade.

Midterm Exam: Median score51, Mean 60.36

*Thanks to Kevin Scannell for these links. Students are welcome to suggest others.

This page is maintained by Robert Hardt ( email)

Last edited 12/02/03..