Math 120A Differential Geometry

Lect.: MWF 10-10:50 in MS 5138
Disc.: Th 10-10:50 in MS 5138

Office hours: M 2-2:30, W 11-12:30,
F 11-11:30 in MS 5366

Teaching Assistant: Danielle O'Donnoll (daodonno@math.ucla.edu)

Office hours: TBA in MS 2954

Course information

Course outline: We will study differential geometry of regular curves and surfaces in three-dimensional space. The course assumes excellent command of mathematial analysis and linear algbera.
We will begin with local properties of regular curves: curvature (which measures how much the curve bends) and torsion (measuring how much the curve twists). We will prove that these two quantities, considered as functions of the parameter on the curve, determine a curve up to rigid motion and reparametrization. Some global properties of curves will also be developed (e.g., we will prove that a closed curve of a given length which bounds the largest area is the circle).
The main goal of the course is to generalize as much as possible the picture for the curves to the case of two-dimensional surfaces. We will learn how to measure lengths of curves, angles between curves, areas of regions, etc., on a regular surface. We will also study various notions of curvature on a surface, which come up in different problems of differential geometry.
Finally, we will study isometries (distance preserving maps), geodesics (the shortest paths between two given points on a surface), etc.  The course ends with the celebrated Gauss-Bonnet theorem, which, roughly speaking, relates the sum of the angles of a triangle on a two-dimensional surface to the curvature of the surfaces.

Textbook:
Manfredo
P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall.

HOMEWORK: Homework is an essential part of the course, since trying to solve a lot of different problems on your own is the only way to learn how to come up with proofs and write them down. Homework will be assigned weekly and will be collected on Friday in class.  Several problems, chosen at random, will be graded each week. However, it is recommended to try to solve each of the assigned problems. Two lowest HW scores will be dropped.

EXAMS AND QUIZZES: Midterms: February 4th (Wednesday) and March 3rd (Wednesday), in class. Final: March, 22nd (Monday), 8a.m.-11a.m.. There will be no make-up exams. Throughout the semester, there will be several quizzes in the discussion section. Quiz dates and topics will be announced in advance. One lowest quiz score will be dropped.

• HW (15%) + Midterm1(15%)+Midterm2(15%)+Quizzes(15%)+Final (40%);
• HW (15%) + BestMidtermScore(15%)+ Quizzes(15%) + Final (55%)
Number
Material
Homework
Notes
HW 1 due 1/16
1.2-1.4
1.2: 1-4; 1.3: 1, 2, 5, 6, 10; 1.4: 1-3

HW 2 due 1/23
1.4, 1.5
1.4: 9,10, 11a), 5 (e.g., use 11), 12; 1.5: 1, 3, 6

HW 3 due 1/31
1.5, 1.7
1.5: 5, 7, 8, 12a)b)d), 13; 1.7: 3, 6, 8, 10, 12

HW 4 due 2/06
TBA

Midterm 1: Feb. 4ths
Practice Midterm (PDF)
HW 5 due 2/13
2.2, 2.3
2.2: 1-3, 6, 7, 10, 12, 16; 2.3: 4, 7, 10, 15

HW 6 due 2/20
2.4, 2.5
2.4: 1, 2, 8, 15, 20; 2.5: 1a, 3, 5, 11

HW 7 due 2/27
3.2, 3.3
3.2: 1, 3, 5, 6, 8a, 16; 3.3: 1, 2, 6a, 20, 23
Practice Midterm (PDF)
HW 8 due 3/5
TBA

Midterm 2: Mar. 3rd
HW 9 due 3/12
TBA

HW 10 due 3/18 [in section]
TBA

Notes on the Gauss Map and 2nd Fundamental Form
Final Exam (comprehensive): Monday, March 22nd, 8-11am, room TBA