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MATH-F413
Géométrie riemannienne
Titulaire(s) du cours
Joel FINE (Coordonnateur)Crédits ECTS
5
Langue(s) d'enseignement
français
Contenu du cours
This course, which is given in English, goes from the foundations of Riemannian geometry up to the interaction between topology and curvature. The foundational material includes: the definition of a Riemannian metric; the lengths of paths, areas of surfaces and volume in general; the natural derivative on a Riemannian manifold (called the Levi-Civita connection ); the definition of geodesics, which are the analogues of straight lines in the sense that they minimising length; the definition of curvature. From here we explore topics such as Jacobi fields, which describe how geodesics behave in families and which shows how curvature influences this behaviour. From here we move to one of the central themes of the subject: how the topology of a manifold restricts the curvature of the possible Riemannian metrics it can carry.
Objectifs (et/ou acquis d'apprentissages spécifiques)
- Familiarity with the basic concepts: Riemannian metric, Levi-Civita connection, curvature tensor, geodesics
- Ability to calculate with these definitions.
- Understanding of Jacobi fields, and the appearance of curvature in Jacobi's equation
- Appreciation of the interaction between topology and curvature, as formalised in various theorems.
- Ability to calculate and reason based on these above results.
Pré-requis et Co-requis
Cours co-requis
Méthodes d'enseignement et activités d'apprentissages
The course is made up of 24 hours of lectures, which are supplemented by 24 timetabled hours in which the titulaire will be available in his office to answer all questions related to the course. There will be regular exercise sheets which are mandatory although they will not be marked unless the student requests it explicitly.
Support(s) de cours
- Syllabus
Autres renseignements
Contacts
Joel Fine (joel.fine@ulb.be)
Campus
Plaine
Evaluation
Méthode(s) d'évaluation
- Examen écrit
Examen écrit
Construction de la note (en ce compris, la pondération des notes partielles)
The final mark will be based entirely on performance in the written exam.
Langue(s) d'évaluation
- anglais