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- B+ B B- C+ C C- D+ D F
grade 93-100 90-92 87-89 83-86 80-82 77-79 73-76 70-72 67-69 60-66 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

Saint Joseph’s University encourages the free and open pursuit of knowledge; we consider this to be a fundamental principle and strength of a democratic people. To this end, SJU expects its students, its faculty, its administrators, and its staff to uphold the highest standards of academic integrity. The University expects all members of the University community to both honor and protect one another’s individual and collective rights.

Student with Disabilities Statement

Reasonable academic accommodations may be provided to students who submit appropriate documentation of their disability. If students have need of assistance or questions with this issue, they are encouraged to contact the Office of Student Disability Services (SDS) at sds@sju.edu or by phone at 610.660.1774. The Office of SDS also provides an appeal/grievance procedure for complaints regarding requested or offered reasonable accommodations. More information can be found at: www.sju.edu/sds.

Health and Wellness Statement

Saint Joseph's University recognizes that physical and mental health strongly impact one's ability to do well in school and in life. As a result, there are many helpful campus resources designed to help students to care for their physical, mental, and spiritual health. Students may experience stressors that can impact both their academic experience and their personal well-being. These may include academic pressure and challenges associated with relationships, mental health, alcohol or other drugs, identities, finances, etc. All of us benefit from support during times of struggle and challenges. If you are experiencing concerns, seeking assistance sooner rather than later is a courageous thing to do for yourself and those who care about you. The resources at https://sites.sju.edu/counseling/ can help you to cope with stress and to achieve your academic and personal goals.