Differential Geometry of Curves and Surfaces — 1^st Semester — 2024/2025

Annoucements

Homework 8 is due on November 22.

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

Secundary

• do Carmo, Differential Geometry of Curves and Surfaces
• O'Neill, Elementary Differential Geometry

Grading Policy

Homework: Makes up 50% of the grade. Late homework will not be accepted.

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

Additional Resources

Abbreviated lecture notes

Differentiable manifolds and differential forms

Interesting links: Foucault pendulum, Gravity Probe B, Minimal Surfaces, Helicoid to Catenoid.

Homework

• Week 1 (due on September 20)
• Week 2 (due on September 27)
• Week 3 (due on October 4)
• Week 4 (due on October 11)
• Week 5 (due on October 18)
• Week 6 (due on October 25)
• Week 7 (due on November 15)
• Week 8 (due on November 22)

Exams

• Exam 1

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

• 2023/2024