In this unit we develop the differential geometry of curves and surfaces, up to the point where the student is ready to begin a study of abstract manifolds and riemannian geometry.
At the end of the course the student should be able to prove the essential results of the geometry of curves and surfaces and as well as master the techniques for calculating the geometrical objects in specific examples.
1. Curves in R3: Frenet referential, surfaces in R3, tangent space, orientation, Gauss map, curvature, invariants of a surface;
2. Intrinsic geometry: vector fields, covariant derivative, geodesics;
3. Differentiable manifolds (essentials)
4. Riemannian varieties (essentials)
Neto, O., Tópicos de Geometria, Universidade Aberta, 1999. https://repositorioaberto.uab.pt/handle/10400.2/13517
Manfredo do Carmo, Geometria Diferencial de Curvas e Superfícies, 4th ed., Rio de Janeiro, SBM, 2010.
Continuous assessment is privileged: 2 or 3 digital written documents (e-folios) during the semester (40%) and a presence-based final exam (p-folio) in the end of the semester (60%). In due time, students can alternatively choose to perform one final presence-based exam (100%).
Students are recommended to have previously knowledge in Linear Algebra I, Geometry and Elements of Infinitesimal Analysis I, II and III.