2025
Topics of Differential Geometry and Topology
Name: Topics of Differential Geometry and Topology
Code: MAT11700D
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese
Regime de Frequência: Presencial
Sustainable Development Goals
Learning Goals
Abstract thought, logical deductive reasoning, formalization and mathematical rigour, etc.
Initiate the study of Differential Geometry, promoting interest and skills amongst students willing to do research in Riemannian Geometry or the Topology of Manifolds.
Initiate the study of Differential Geometry, promoting interest and skills amongst students willing to do research in Riemannian Geometry or the Topology of Manifolds.
Contents
Part 1 - Metric and topological spaces. Fundamental group. Covering space, universal covering space. Examples and applications.
Part 2 - Differentiable manifolds and the tangent bundle. Vector fields and orientation. Manifold-with-boundary and induced orientation. Transformations between manifolds. Brief notions of submanifold theory. Differential forms, exterior derivative. De Rham cohomology, Poincaré lemma. Integration on manifolds and Stokes theorem.
Part 3 - Riemannian manifolds and volume. Geodesics, Riemannian parallel transport. Curvature and the holonomy group. Vector bundles. Natural fibre bundles over a manifold.
Part 4 - Further topological notions, the Euler characteristic. Lie groups and álgebras. Lie group actions on manifolds. Singular homology.
Part 2 - Differentiable manifolds and the tangent bundle. Vector fields and orientation. Manifold-with-boundary and induced orientation. Transformations between manifolds. Brief notions of submanifold theory. Differential forms, exterior derivative. De Rham cohomology, Poincaré lemma. Integration on manifolds and Stokes theorem.
Part 3 - Riemannian manifolds and volume. Geodesics, Riemannian parallel transport. Curvature and the holonomy group. Vector bundles. Natural fibre bundles over a manifold.
Part 4 - Further topological notions, the Euler characteristic. Lie groups and álgebras. Lie group actions on manifolds. Singular homology.
Teaching Methods
Structured exposition, examples, exercices, home-work assignments - these are used for evaluation. Evaluation by an extra work or exam if required.
Teaching Staff
- Rui Pedro Lima Pinto Ribeiro de Albuquerque [responsible]
Certificações
