Math 120B Differential Geometry

MWF 10-11 MS5217

Instructor: Olga Radko (radko@math.ucla.edu)

Office hours: MW 3-4 and by appointment

Note: This course is a continuation of the first quarter of differential geometry. The knowledge of the material covered in math120a is essential for the understanding of math 120b. If you did not take 120a, approval of the instructor is necessary to register.

Course description.

                The course starts with basic properties of  vector fields;  applications and corollaries of the Gauss-Bonnet theorem; covariant derivatives and parallel transport in relation to geodesics;  the exponential map and the geodesic polar coordinates. The study of  intrinsic geometry of surfaces is concluded by the fundamental theorem of local theory of surfaces.
                 After this, we will consider several results on the global differential geometry of curves and surfaces (e.g., Fenchel, Fay-Milnor, Jordan Curve theorems; the rigidity of the sphere, Hopf-Rinow theorem, etc.)
                 The last part of the course will be devoted to introduction to general manifolds. We will start with tensor algebra, and proceed to vector analysis and integration on manifolds, with the basics of de Rham cohomology being the main goal.

The main textbook for the first two parts of the course will be "Differential geometry of  curves and surfaces" by M. do Carmo. Additional texts will be distributed as necessary.


 

Grading policy

The grading scheme for this class will be as follows:

Attendance and Class participation (10%) + Homework (10%)+Midterm(20%)+Project(30%)+Final(30%)

Final exam solutions are now available.

Class Project

As a part of this class, you will be asked to write a short survey/research paper or make in-class presentation on a topic related to the class material. We will start discuss the possible topics shortly.

Project Topics

Name
 Presentation Title
Eugene Chen
 3-body problem in classical mechanics
Jordan Fassler
 Minimal surfaces
Ivan Ip
 Lie derivatives and Cartan's  magic formula
Marcos Morinigo
 Non-Euclidean  Geometries
Yuan Xu
 Bonnet's theorem
Faviola Arroyo
 TBA
Ho Ng
 TBA


Homework

There will be regular (approximately, biweekly) homework, which will be announced in class, and send by e-mail through my.ucla.edu to all registered students.