MA 430 Differential Geometry


handouts syllabus pdf flier photographs

Fall 2021

The classes will be held on campus. Mon 2:00–3:50 pm in STC 148, and Fri 1:00–1:50 pm in STC 337.

Office Hours

There will be two types of office hours:
1. In-person for short (15 min or less) questions, one student at a time, no appointment necessary on: Mondays at 1 pm and 4 pm, on Thursdays at 4 pm, and on Fridays at 2 pm.
2. Online for any format (short or long questions, individual or in groups), by appointment: email me and I will find time for us to meet in a D2L Brightspace room. During these meetings, I will have a board behind me and you will be able to see what I write on it while I answer your questions. So, this format will be very similar to in-person meetings in my office.


No textbook is required. Handouts with new material and practice problems are available on my website for each topic. The textbooks used for the class preparation include the following:
Richard Millman, George Parker, Elements of Differential Geometry
Richard L. Faber, Differential Geometry and Relativity Theory

Topics Covered

1. Curves: parametrization, tangent vector, arc length, acceleration vector, curvature, normal and binormal vector, torsion, Frenet-Serret apparatus
2. Surfaces: tangent plane, curvature, Theorema Egregium.
3. Surfaces: coordinate patches, the First Fundamental Form.
4. Surfaces: the Second Fundamental Form, the Gauss curvature, geodesics, curvature tensor, manifolds.


Exam 126%
Exam 226%
Exam 326%
Grades are computed according to the following system:
A+ A A- B+ B B- C+ C C- D+ D D- F
grade 97-100 93-96 90-92 87-89 83-86 80-82 77-79 73-76 70-72 67-69 63-66 60-62 0-59

Number of credits



Calculus 3 or permission of instructor.


The class lectures will be delivered on campus (at the times listed above). To stay on track, it is highly recommended that students attend all classes. I will be available to any student who misses a class and has questions.
Students are responsible for all material covered in class, even if attendance is not checked or assignments collected.

Exams and Projects

There will be three exams and one project during the semester. Project or exams turned in after their due date will receive an automatic reduction in grade. No exam grade will be dropped.

Response time

The exams and project are typically graded in three days after they are turned in. Special circumstances like snow days, school closing or holidays, may occasionally delay the response time. Barring special circumstances, students’ emails are usually responded to within one working day.

More on the course topics

  • The course will start with a review of Multivariable Calculus. The course can be considered a continuation of Calculus 3 course and the next step in deepening the students understanding of calculus and building students' problem solving skills.
  • The review of multivariable calculus will morph into the study of differential geometry - a mathematical discipline that uses methods of multivariable calculus to study geometrical features, such as shape and curvature, of objects. The curvature measures the extend of bending of a curve, a surface, a space or their generalizations to any dimension, the manifolds. Studying ways of describing such an extent of bending is one of the central ideas of the course and enables one to understand concepts like the expansion rate of the universe.
  • Differential Geometry is used in natural sciences, especially in physics and computational chemistry.
  • The course provides the students interested in continuing their education at a graduate level with mathematical techniques that certain graduate programs use.

Course Objectives

  • Identify situations that require the use of vector calculus and differential geometry.
  • Solve certain classes of problems related to vector calculus, differential geometry or topology.
  • Understand and write mathematical proofs using formal mathematical reasoning.
  • Present solutions on computer or in a written form.

Learning outcomes

Students will:
  • develop understanding of basics of differential geometry,
  • be able to understand and solve problems which require the use of differential geometry,
  • know how to use formal mathematical reasoning and write mathematical proofs when necessary,
  • demonstrate ability to cover a topic independently and to present their results in a written report.

Academic integrity

Academic integrity is at the center of the educational experience at USciences. Students are therefore expected to uphold the highest standards of academic integrity and not engage in nor tolerate academic dishonesty. Academic dishonesty includes, but is not limited to, fabrication, cheating or plagiarism, and unauthorized collaboration. Any violation of academic integrity will be investigated and, where warranted, the student will receive appropriate sanctions through the University's Student Conduct Process. Please familiarize yourself with the current USciences Student Handbook. In particular, adherence to the Student Conduct Policy and Academic Integrity Policy will help to ensure that your learning and living experiences are founded on integrity.

Americans with Disabilities Act (ADA) Compliance Statement

USciences supports the educational endeavors of all students, including students with disabilities. ADA defines a disability as a mental or physical impairment that substantially limits one or more major life activities. If you believe that you have a disability that may impact your ability to fulfill your course or degree requirements, and you would like more information on applying for an accommodation under ADA, please contact the Program Coordinator of Student Accommodations who serves as 504 Coordinator at 215-596-8758 or

Mental Health Wellness Statement

USciences encourages students to recognize that academic success requires students to be emotionally and physically well. If you are having difficulty coping with stress associated with the classroom or are experiencing other personal issues, please go to USciences Health and Counseling (SHAC) located on the first floor of Whitecar Hall, or call 215.596.8536, or email Additional emotional support is available 24/7 and can be obtained by contacting the National Suicide Prevention Hotline at 800.273.8255 or by texting “Go” to the Crisis Text Line 741-741. The services listed above are all free and confidential.

COVID-19 Syllabus Statement

As we continue to navigate the COVID-19 pandemic it is important to remember that:
· All individuals, regardless of vaccination status, are required to wear masks indoors on campus.
· USciences mandates that all students, faculty, and staff are vaccinated against COVID-19.
· Practice good hygiene by frequently washing hands.
· Conduct a daily health check before coming to campus.
Check the COVID-19 Return to Campus Guidelines regularly for updates. If we all follow the rules, we will help to ensure that our learning environment is as safe as possible.
Students who are ill, regardless of symptoms, vaccination status or or diagnosis, should not come to campus for classes or activities and should contact SHAC. If students are ill, but able to virtually participate in courses they may do so. Students should not contact the Office of Student Health for “sick notes.”
The COVID-19 pandemic has refocused the need for both mental and physical personal wellness. The University encourages students who are ill to take the time to focus on their health. Additionally, if needed students may reach out to their college dean’s office to inquire about leave of absence options including: short-term leave of absence (up to 10 business days), a personal leave of absence, or a medical leave absence. We firmly believe to learn you must be healthy.