Differential Geometry of Curves and Surfaces — 1^st Semester — 2025/2026

Lecturer:	José Natário

	Email: jnatar@math.tecnico.ulisboa.pt

	Office: Mathematics Building, 4th floor, room 4.29

Classes:	Mondays from 14:00 to 16:00 in room E5, and Fridays from 8:30 to 10:00 in room E1
Office Hours:	Drop by my office or send me an email

Annoucements

Homework 11 is due on December 5.

You can watch an unstable soap catenoid collapsing into two disks here (explanation here). Notice how the extra area is used to form a small bubble.

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.