Riemannian Geometry

Code: MAT14349M

6 ECTS

Duration: 15 weeks/156 hours

Scientific Area: Mathematics

Teaching languages: Portuguese

Languages of tutoring support: Portuguese

Presentation

The aim of this CU are the very complete understanding by the student of the notion of variety, Riemannian varieties and their properties, as well as the notion of tangent fibre, affine connection, torsion and curvature.

Sustainable Development Goals

Educação de qualidade - Assegurar a educação inclusiva, e equitativa e de qualidade, e promover oportunidades de aprendizagem ao longo da vida para todos.

Learning Goals

The intended outcomes of this UC are the students very complete understanding of the notion of manifold, Riemannian manifold and its properties, as well as the notion of tangent bundle, affine connection, torsion and curvature. Starting form abstaract manifolds, it intends to arrive at the Riemannian geometry of isometrically immersed submanifolds, passing through some elements of manifold topology and manifolds-with-boundary. The learning outcomes shall be the knowledge of the core theory of Riemannian geometry and some classical applications, such as the theory of surfaces in R3, and the acknowledgement of further related research fields.
It is intended to enrich the mathematical knowledge of students with the last centurys main geometry ideas, which have persisted and have been renewed in recent issues, which students may want to continue exploring e.g. in a doctoral program.

- Manifolds: tangent bundle, vector field, Lie bracket, immersion and embbeding, orientation, Lie groups, manifold topology, manifold with boundary.
- Riemannian manifolds: metric, isometry, Riemannian submanifold, affine connection, torsion, Levi-Civita connection, geodesics and normal charts, completeness and the theorem of Hopf-Rinow.
- Curvature: curvature tensor, sectional curvature, Ricci tensor, scalar curvature, space-forms, Jacobi fields, theorems of Myers and Hadamard.
- Study of surfaces: Gauss map, second fundamental form, mean curvature, theorem of Gauss.

Teaching Methods

This course will follow as a natural development of a course on Differential Geometry.
Admitting the estudents do not possess such a base, the professor shall orient himself by the average of the class, starting the course well on the theme of manifolds and taking the students as far as he can within the program. On the other hand, if the students have a previous knowledge of manifolds, then the course shall start with Riemannian metrics and certainly continue until the end of the program or further on.
The teaching method will be of the classical type, with theory exposition and by solving proposed problems.
Continuous evaluation, for two to three frequencies or oral presentations of topics to be agreed with the teacher, totaling 80% of the classification (number of presentations to be defined by the responsible teacher, taking into account the characteristics of the students and the plan of classes) and homework, totaling 20%; or by final exam.