Math 42

Winter 2022

Differential Geometry I

General Information








Vladimir Chernov

Contact Information

Office: 304 Kemeny Hall

Telephone: (603) 646 2421


Time and classroom

Course Meeting times:

Monday, Wednesday, Friday 8:50-9:55 AM, x-hour Thursday 9:05-9:55 AM in Kemeny 105

Office Hours:

Tuesday 3-5 PM, Friday 3-4 PM



Text Book

John Oprea “Differential Geometry and its Applications”, second edition

Course grade will be based on



Three two-week homework sets that are due on Wednesdays January 19, February 2 and March 2 at noon. Homework will be 30% of the total grade.

Late homework will not be accepted unless it is excused in advance for serious reasons. Unexcused late and missing homework counts zero.

Middle of the Term Presentation and Discussion to be held Sunday February 6 – Saturday February 12. The middle of the term presentation and discussion is 30% of the total course grade.

End of the Term Presentation and Discussion to be held Wednesday March 9 – Saturday March 12. The End of the Term presentation and discussion is 40% of the total course grade.



ORC Course Description: This course will cover curves and surfaces in Euclidean 3-dimensiona space. Topics include curvature and torsion of curves, the Frenet-Serret equations, Gaussian and mean curvature of surfaces, geodesics and parallel transport, isometries and Gauss’s Theorem Egregium, the Riemann curvature tensor. One or more of the following topics will be studied if time permits: vector fields, tangent bundles, hypersurfaces, connections and curvatures. Offered in alternate years.


Prerequisites:  Math 22 and Math 23 or the permission of the instructor




Honors Code: Collaboration on homework is allowed and in fact encouraged but the final solutions should be written by each individual student. There is no help given or received allowed during the Middle and End of the Term presentation and discussion. You are of course allowed to bring the materials needed for the presentation but no material to help you with the discussion portion will be allowed.

Tentative Course Schedule: Weeks 1 and 2: The Geometry of Curves, Weeks 3 and 4: Surfaces, Weeks 5 and 6: Curvatures, Weeks 7 and 8: Constant Mean Curvature Surfaces, and Geodesics, Week 9: Gauss theorem, Toponogov theorem briefly.


Attendance: You are expected to attend class in person unless you have made alternative arrangements due to illness, medical reasons, or the need to isolate due to COVID-19.

For the health and safety of our class community, please: do not attend class when you are sick, nor when you have been instructed by Student Health Services to stay home.

Safety: In accordance with current College policy, all members of the Dartmouth community are required to wear a suitable face covering when indoors, regardless of vaccination status. This includes our classroom and other course-related locations, such as labs, studios, and office hours. If you need to take a quick drink during class, please dip your mask briefly for each sip. Eating is never permitted in the classroom. (The only exception to the mask requirement is for students with an approved disability-related accommodation; see below.) If you do not have an accommodation and refuse to comply with masking or other safety protocols, I am obligated to assure that the Covid health and safety standards are followed, and you will be asked to leave the classroom. You remain subject to course attendance policies, and dismissal from class will result in an unexcused absence. If you refuse to comply with masking or other safety protocols, and to ensure the health and safety of our community, I am obligated to report you to the Dean’s office for disciplinary action under Dartmouth’s Standards of Conduct. Additional COVID-19 protocols may emerge. Pay attention to emails from the senior administrators at the College.


If needed we may have to move the course to ZOOM due to COVID or other related situations.

Consent to Recording: (1) Consent to recording of course and group office hours a) I affirm my understanding that this course and any associated group meetings involving students and the instructor, including but not limited to scheduled and ad hoc office hours and other consultations, may be recorded within any digital platform used to offer remote instruction for this course; b) I further affirm that the instructor owns the copyright to their instructional materials, of which these recordings constitute a part, and distribution of any of these recordings in whole or in part without prior written consent of the instructor may be subject to discipline by Dartmouth up to and including expulsion; b) I authorize Dartmouth and anyone acting on behalf of Dartmouth to record my participation and appearance in any medium, and to use my name, likeness, and voice in connection with such recording; and c) I authorize Dartmouth and anyone acting on behalf of Dartmouth to use, reproduce, or distribute such recording without restrictions or limitation for any educational purpose deemed appropriate by Dartmouth and anyone acting on behalf of Dartmouth. (2) Requirement of consent to one-on-one recordings By enrolling in this course, I hereby affirm that I will not under any circumstance make a recording in any medium of any one-on-one meeting with the instructor without obtaining the prior written consent of all those participating, and I understand that if I violate this prohibition, I will be subject to discipline by Dartmouth up to and including expulsion, as well as any other civil or criminal penalties under applicable law.


Religious Observances: Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with me as soon as possible, or before the end of the second week of the term—at the latest, to discuss appropriate adjustments. Dartmouth has a deep commitment to support students’ religious observances and diverse faith practices.


Student Accessibility Services: Students requesting disability-related accommodations and services for this course are required to register with Student Accessibility Services (SAS; Getting Started with SAS webpage; and to request that an accommodation email be sent to me in advance of the need for an accommodation. Then, students should schedule a follow-up meeting with me to determine relevant details such as what role SAS or its Testing Center may play in accommodation implementation. This process works best for everyone when completed as early in the quarter as possible. If students have questions about whether they are eligible for accommodations or have concerns about the implementation of their accommodations, they should contact the SAS office. All inquiries and discussions will remain confidential.


Mental Health Even without the global pandemic, the academic environment at Dartmouth is challenging, our terms are intensive, and classes are not the only demanding part of your life. In the midst of a global pandemic, with all the uncertainty surrounding every aspect of our lives, these challenges take on an extra toll. There are a number of resources available to you on campus to support your wellness, including your undergraduate deanCounseling and Human Development, and the Student Wellness Center.