# MATH 401: Differential Geometry, Fall 2003

### Description:

Differential geometry is the study of geometric figures using the methods of calculus.  It has a rich history.  See the brief biographies in the links to Some Classical Geometers below. With origins in cartography, it now has many applications in various physical sciences, e.g., solid mechanics, computer tomography, or general relativity. In this course we develop much of the language and many of the basic concepts of differential geometry by consideration of curves and surfaces in ordinary3 dimensional Euclidean space. Some of the most striking results in geometry involve relations between local concepts (e.g. the index of a vectorfield or the curvature of a surface at a point) with a global quantity such as the number of handles of a closed surface.  For example the Poincare-Hopf theorem implies that any continuous vectorfield tangent to the sphere must vanish somewhere, or the Gauss-Bonnet theorem implies that the integral of the curvature over any topological donut, however lumpy, must be zero.

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
Prerequisites for the course include some familiarity with calculus, linear algebra, and ordinary differential equations

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 1.   A Fixed Point Theorem
Handout 2.   A Very Short Course in Local O.D.E. Theory
Handout 3.   TheFundamental Theorem of Space Curves
Handout 4.   The Implicit and Inverse Function Theorems

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) .

### Sections to be (tentatively) covered in Do Carmo:

Chapter 1 :  1.1, 1.2, 1.3, 1.4, 1.5, 1.7ABC
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

### Other elementary books:

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.

Math 402, Spring 2004:
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

• Beltrami
• Blaschke
• Bonnet
• Darboux
• Frenet
• Gauss
• Klein
• Lobachevsky
• Monge
• Riemann