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# MTH 255 - Differential Geometry

### Course Instructor

• Name: Ustun Yildirim
• Email: ustun.yildirim (a t) rochester (d o t) edu
• Office: Hylan 1001
• Office hours:
• Mon 4pm - 5pm
• Tue 12:30pm - 1:30pm

Classroom & Time : MW 12:30-1:45 pm in Hylan 306

Prerequisites: MTH164 and MTH235, or MTH174

Textbook: Differential Geometry of Curves and Surfaces By Manfredo P. Do Carmo

References:

1. A Comprehensive Introduction to Differential Geometry by Mike Spivak vol 2 and 3.
2. Calculus on Manifolds by Michael Spivak
3. Elementary Differential Geometry (2nd edition), by Barett O’Neill.
4. Everything else you can find about the subject!

### Description & Syllabus

This course uses mutli-variable calculus and linear algebra to study geometry of curves and surfaces. The key concept introduced here is curvature, a sort of second order derivative quantity. The storyline of this course is that we first develop local theory on defining curvatures, then we prove existence and uniqueness results, finally the course culminated in proving several amazing theorems linking local geometric quatities to global geometrical or topological information of curves and surfaces.

The following is the outine of the course.

Chapter 1: Curves, Torsion, Curvature, Frenet, Fundamental theorem of the local theory of curves, Global properties of plane curves.

Chapter 2: Surfaces, Tangent plane, Differential of a map, First fundamental form, Area

Chapter 3: Gauss map, Second fundamental form, Vector fields, Ruled surfaces and minimal surfaces

Chapter 4: Isometries, Conformal maps, Gauss theorem, Gauss-Bonnet theorem

There will be one midterm, one final and 7 homework sets.

Midterm will be on Wednesday, October 23, 2019 during class.

Final exam will be on Monday December 16, 2019 at 7:15pm-9:15pm.

If you need to take a make up exam, you have to contact the instructor as soon as possible. Make up exams will only be offered for valid reasons with proper documentation.

Homework sets will be due every two weeks on Wednesdays at the beginning of the class. They will be posted here.

If you cannot make it to the class to turn in a homework, please bring it to my office (you may slip it under my door) before the deadline.

• Midterm: 20%
• Final: 30%
• Homework: 50%
• Your lowest homework set will be dropped.

### Disability Support

The University of Rochester respects and welcomes students of all backgrounds and abilities. In the event you encounter any barrier(s) to full participation in this course due to the impact of disability, please contact the Office of Disability Resources. The access coordinators in the Office of Disability Resources can meet with you to discuss the barriers you are experiencing and explain the eligibility process for establishing academic accommodations. You can reach the Office of Disability Resources at: (585) 275-3424; Taylor Hall. Or, visit Disability Resources.

Please note that to be granted alternate testing accommodations, you (the student) must fill out forms with Disability Resources at least seven days before each and every exam. These forms are not sent automatically. Instructors are not responsible for requesting alternative testing accommodations at CETL or making accommodations on their own.