MATH 436 -- Differential Geometry of Curves and Surfaces

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Starting with multi-variable calculus, this course will develop the theme of invariants attached to the intrinsic and extrinsic geometry of curves and surfaces. Using local coordinates, invariants will be defined, which will later turn out to be independent of the choice of coordinates. The contrasts between intrinsic and extrinsic concepts will be emphasized. The notion of a smooth submanifold will be explored in detail, as will various notions of curvature. The various notions of curvature of surfaces are related to curvature and torsion of curves. The contrast between local and global phenomena is also emphasized. In the past the course has dealt with surfaces of revolution, ruled surfaces, minimal surfaces, special curves on surfaces, Gauss's "Theorema Egregium" and the Gauss-Bonnet theorem.

Here is some more basic information about MATH436:

Time & Place:

TuTh 11:00-12:15 pm, Math B0421

Instructor:

Professor Richard A. Wentworth
Office: 3109 Mathematics Building
Phone: (301) 405-5130
Office Hours: Tu-Th 9:30-10:30 am, and by appointment
Email: raw@umd.edu
Web: www.math.umd.edu/~raw

Text:

Elementary Differential Geometry, First Edition, by Andrew Pressley, Springer Undergraduate Mathematics Series, ISBN 1-85233152-6 (2001)

Homework:

There will be periodic homework assignments. In addition, you should work through the problems in the text, since they are typical of the types of problems that will appear on the exams.

Exams:

There will be three midterm exams on Sep 25 , Oct 25 , and Nov 29 . These will be given during the normal lecture period in the usual room. There will be a comprehensive final exam on Dec 13, 8:00-10:00 am. More information will be posted here: FINAL EXAM INFO.

Makeups:

There will be no makeups for midterms. If you have compelling reasons for missing an exam let me know as soon as possible.

Grading:

The final grade will depend on your performance on the exams and homework. The breakdown is as follows: midterm exams = 100 points each, final exam = 200, homework = 100.

Academic Integrity :

You should be familiar with the University's policies on Academic Integrity, including the Honor Pledge.

Students with disabilities :

If you have some disability related to testing under the usual timed, in-class conditions, you may contact the office of Disabled Students Services (DSS) in Shoemaker. If they assess you as meriting private conditions and/or extra time, then you may arrange to take your tests at DSS, with extra time as they indicate. You must arrange this well in advance of a test. Click to Disability Support Services for further information.

Religious observances:

If your religion dictates that you cannot take an exam or hand in assigned work on a particular date, then contact me at the beginning of the semester to discuss alternatives. You are responsible for making these arrangements.

Detailed Syllabus:

Below is an outline of the material I hope to cover. This will change often as the semester progresses, so check here often for updates. The reading selections are from Pressley.

WEEK

DATE

TOPICS

READING

HOMEWORK ASSIGNMENTS

1

Aug 30

Introduction. Curves.

Ch. 1

2

Sep 4

Curves and curvature.

Ch. 2

3

Sep 11

Frenet-Serret formula.

Ch. 2-3

4

Sep 18

Umlaufsatz & Four vertex theorem.

Ch. 3

5

Sep 25

First exam. Surfaces.

Ch. 4-5

6

Oct 2

Examples. Inverse function theorem.

Ch. 5-6

7

Oct 9

First fundamental form.

Ch. 6

8

Oct 16

Curvature and the second fundamental form.

Ch. 7

9

Oct 23

Second midterm.

Ch. 7-8

10

Oct 30

Gauss, mean, and principal curvatures.

Ch. 8

11

Nov 6

Curvature (cont.)

Ch. 8-9

12

Nov 13

Geodesics.
Ch. 9

13

Nov 20

Theorema Egregrium.

Ch. 10

14

Nov 27

Theorema Egregrium (cont.) Gauss-Bonnet. Third exam.

Ch. 10,13

15

Dec 4

Gauss-Bonnet (cont.)

Ch. 13

16

Dec 11

Review.

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Richard A. Wentworth