Geometria Riemanniana

Objectives

Familiarize the students with the fundamental concepts of differential geometry. Introduce the language and the basic results of Riemannian geometry, with special emphasis on surfaces in three dimensional Euclidean space.

Syllabus

Differentiable Manifolds: Tangent space; differentiable maps; immersions and embeddings; vector fields, Lie brackets and flows; Lie groups; orientability; manifolds with boundary. Differential forms: tensor fields; differential forms; exterior derivative; integration of differential forms; Stokes theorem; orientation and volume forms. Riemannian Manifolds: isometries; affine connections, Levi-Civita connection; geodesics, minimizing properties of geodesics; Hopf-Rinow theorem. Curvature: curvature tensor, sectional curvature, Ricci tensor, scalar curvature; connection and curvature forms, Cartan's structure equations; index of a vector field at a singularity, Euler- Poincaré characteristic, Gauss-Bonnet theorem; isometric immersions, second fundamental form, mean and Gaussian curvatures.

Prerequisites

Differential Geometry of Curves and Surfaces.

Cross Competence Component

The UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.

Ethical Principles

All members of a group are responsible for the group’s work. In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.