Office Hours: TBA and by appointment

Office Hours: TBA and by appointment

Course Description: This course is an introduction to differential geometry, studying the geometry of curves and surfaces. Some of the key notions discussed include curvature (measuring how much they bend), geodesics (the shortest path between two nearby points) and parallel transport, surfaces of constant curvature and minimal surfaces (surfaces minimizing area in a certain sense, eg soap bubbles). Some gems along the way are the Gauss-Bonnet theorem (a beautiful theorem relating the geometry and topology of a surface) and Stokes' theorem (a central ingredient in many applications, which generalizes the Divergence Theorem, Green’s theorem and Stokes’ theorem from multivariable calculus).

This is an upper division class in mathematics, so there will be a reasonable number of proofs. However, the proofs should merely be a formalization of your visual thinking and intuition. While our main focus will be on concepts and understanding the connections between them, there might be more hands-on computations than in a typical upper division math class, and computer software is recommended for visualization as needed.

Prerequisites: Math 52 or equivalent, and in particular a strong foundation in multivariable calculus, including integration and partial derivatives (and basic linear algebra) which we will use to study the geometry of curves and surfaces.

Course Textbook and Resources: The course does not have a textbook per se, but there are two recommended readings: Neil Donaldson, Introduction to Differential Geometry lecture notes, and Ted Shifrin, Differential Geometry: A First Course in Curves and Surfaces which provides a complementary perspective. (Both are available online and in the Files/Course Materials section of Canvas.)
The course will follow these references, but rather loosely. For this reason, attendance, and taking notes in classes, is strongly encouraged.
A more classical reference you might find useful for some of the topics is Manfredo do Carmo, Differential Geometry of Curves and Surfaces.

Course Policies: Please see https://goto.stanford.edu/mathcoursepolicies for important course policies on exam conflicts, academic accommodations, AI guidance, and taking exams. It is your responsibility to thoroughly read this information.

Course Logistics: Course announcements, homework and other course materials will be posted on Canvas. Weekly homework will be due and graded on Gradescope.

Homework: Weekly homework assignments given out on Wednesday, and due the following Wednesday at 11:59PM on Gradescope (unless otherwise noted). No late submissions will be accepted under any circumstances. Usually only a portion of each week's assigned problems are graded (and the selection of problems chosen to be graded will not be announced in advance).

You are encouraged to attend office hours and to form study groups to discuss and work on homework together, ideally after thinking about it on your own. However, you must write up your own solutions individually and in your own words, and indicate the names of any collaborators or sources of outside help that you received. Keep in mind that simply copying solutions from another student or from other sources (such as AI generated) and then submitting it for credit will be considered a violation of the Stanford Honor Code.

Exams: There will be two in-person, proctored exams: a midterm and a final exam. The exam dates are given below; it is your responsibility to verify right now that you can attend these exams. If you have an academic or a competition-related conflict with the scheduled exam time, please contact us as soon as possible, but no later than two weeks before the exam. If an emergency occurs and you need to miss an exam, contact us as soon as possible.

Homework: 15% (includes some in person assessment)
Midterm Exam: 35%
Final Exam: 50%

• Final Study List deadline: Fri, Oct 9, 5pm (last day to add/drop a class)
• Midterm Exam: Tue, Oct 27, 10:30-11:50am, room TBA.
• Democracy Day: Tue, Nov 3 (No class).
• Thanksgiving Recess: Mon, Nov 23-Fri, Nov 27 (No classes)
• Final Exam: Th, Dec 10, 3:30-6:30pm, room TBA.

• Week 1: Brief intro; Parameterized curves, arc-length (S§1.1 or D§1.1-§1.2)
  Frenet frame, curvature and torsion, Fundamental Theorem of Space Curves (D§1.3-1.4 or S§1.2)
• Week 2: Vector fields, tangent space; 1-forms and differential of a function; line integrals (D§2.1-§2.2)
  Parameterized surfaces, differentials, implicitly defined surfaces (D§3.1 or S§2.1)
• Week 3: First and second fundamental forms, curves in surfaces (D§3.2-§3.3 or S§2.2)
  Gauss and mean curvature, principal curvatures and principal directions (D§3.3)
• Week 4: Euler's formula, elliptic and hyperbolic points (D§3.4 or S§2.2)
  Exterior calculus: wedge product, the exterior derivative (D§2.3)
• Week 5: Gauss’ Theorem Egregium and Fundamental Theorem of Surfaces; Structure equations and adaptive frames (D§3.5 or S§3.3)
  Riemannian geometry, black holes and relativity (D§3.5)
• Week 6: MIDTERM on Tue;
  Geodesics (S§2.4)
• Week 7: Tue: Democracy day (NO CLASS)
  Parallel transport, covariant derivative (S§2.4)
• Week 8: Geodesic curvature; Parallel transport and holonomy; Surfaces of constant curvature (S§3.1 and S§3.3);
  Integration, Stokes’ Theorem
• Week 9: Local Gauss-Bonnet theorem (S§3.1)
  Gauss-Bonnet theorem and applications (S§3.1)
• Week 10: Minimal surfaces and surfaces of constant mean curvature (S§3.4);
  Further topics (TBD)

Academic Integrity: The Honor Code articulates Stanford University's expectations of students and faculty in establishing and maintaining the highest standards in academic work. Its purpose is to uphold a culture of academic honesty. Students will support this culture of academic honesty by neither giving nor accepting unpermitted academic aid in any work that serves as a component of grading or evaluation, including assignments, examinations, and research. Examples of conduct that have been regarded as being in violation of the Honor Code (and are most relevant for this course) include copying from another student's work (or other sources such as the internet, AI generated etc) or allowing another student to copy from your own work; plagiarism; representing as one's own work the work of another. Please visit the OCS website for more information on the Honor Code, and please see link above for the Math Department AI Policy.

Responsibility for understanding: You are expected to understand your submitted work in homework and exams and to be able to verbally explain to your instructor (upon request, and without advance notice) any work you submit. When you apply for jobs, you will be interviewed by professionals who instantly recognize a lack of real understanding due to over-reliance on AI; we hold you to the same standard. Do not submit what you do not understand. The response "I don't understand what I was writing but I know it works'' is never acceptable when asked to explain a submission.