Minimal surface

Differential Geometry of Curves and Surfaces — 1st Semester — 2023/2024

Lecturer: José Natário
Office: Mathematics Building, 4th floor, room 4.29
Classes: Wednesdays from 15:00 to 17:00 in room I9, and Fridays from 9:30 to 11:00 in room V0.06
Office Hours:  Drop by my office or send me an email


Homework 9 is due on November 30.


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.




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



Valid XHTML 1.0! Valid CSS!