Differential Geometry of Curves and Surfaces — 1^st Semester — 2026/2027

Annoucements

Classes start on September 7.

Syllabus

Curves: curves, curvature, torsion, Frenet-Serret formulas, global theorems.

Differentiable manifolds: differentiable manifolds in Rⁿ, tangent space, normal space, parameterizations.

Differential forms: covectors in Rⁿ, exterior product, differential forms, pull-back, exterior derivative, integration, Stokes Theorem.

Surfaces: first and second fundamental forms, mean curvature, Gauss curvature.

Geometry of surfaces: connection and curvature forms, structure equations, Theorema Egregium of Gauss, vector fields, covariant derivative, parallel transport, geodesics.

Gauss-Bonnet Theorem: triangulations, Euler's characteristic, Gauss-Bonnet Theorem.

Minimal surfaces: examples, isothermal coordinates, Weierstrass-Enneper representation.

Biliography

Main

• Kobayashi, Differential Geometry of Curves and Surfaces

Secondary

• do Carmo, Differential Geometry of Curves and Surfaces
• O'Neill, Elementary Differential Geometry
• Pinkall & Gross, Differential Geometry: From Elastic Curves to Willmore Surfaces

Grading Policy

Homework: Makes up 50% of the grade. Students may be asked to solve similar exercises in class.

Final exam: Makes up 50% of the grade. Can be retaken if necessary.

Additional Resources

Abbreviated lecture notes

Differentiable manifolds and differential forms

Lecture notes by former student Tiago Mourão

Interesting links: Foucault Pendulum, Gravity Probe B, Minimal Surfaces, Helicoid to Catenoid, Collapse of an unstable Catenoid.

Homework

• Week 1 (due on September 18)

Exams

• Exam 1

You can see more exams on the course webpages from previous years:

• 2025/2026
• 2024/2025
• 2023/2024