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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


  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!

Course schedule

Lecture notes

Homework and Exercise Sets

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.

Grade distribution is as follows:

  • 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.

Academic Integrity Statement

  • All assignments and activities associated with this course must be performed in accordance with the University of Rochester’s Academic Honesty Policy. More information is available at:
  • In Math 255 calculators, phones, and other electronic devices are strictly forbidden in exams. Notes are also strictly forbidden. You must bring your college id to exams and show it when you hand in an exam.

Course Credit-Hour Policy

This course follows the College credit hour policy for four-credit courses. This course meets 3 academic hours per week. Students may also be expected to deepen their understanding of the course material through close examination/evaluation of the readings assigned in the course.

teaching/fall_2019_mth_255_differential_geometry/main.txt · Last modified: 2019/09/30 13:10 by ustun