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MATH-F310

Differential geometry I

année académique
2025-2026
2026-2027

Titulaire(s) du cours

Andriy Haydys (Coordonnateur)

Crédits ECTS

5

Langue(s) d'enseignement

anglais

Contenu du cours

  • Smooth surfaces ;
  • Differentiation on smooth surfaces ;
  • Partition of unity and integration on smooth surfaces ;
  • Quadratic forms on surfaces ;
  • Abstracts smooth manifolds and smooth maps ;
  • Immersions, embeddings, submanifolds and Whitney’s theorem ;
  • Tangent and cotangent bundles, vector bundles ;
  • Vector fields and 1-parameter groups of diffeomorphisms ;

Objectifs (et/ou acquis d'apprentissages spécifiques)

This course is the first part of the Differential Geometry course. In particular, basic notions and methods of differential geometry such as smooth manifolds, vector fields, vector bundles etc. appearing both in various branches of mathematics and physics will be introduced and developed. At the end of this teaching unit, a student will be able 
•    to decide whether a given topological space is a manifold ;
•    to compute the differential of a smooth maps, its critical points ;
•    to compute examples of integral curves of vector fields ;
•    describe properties of 1-parameter groups of diffeomorphisms generated by  vector fields ;

Pré-requis et Co-requis

Connaissances et compétences pré-requises ou co-requises

MATH-F201    Calcul différentiel et intégral II
MATH-F211    Topologie

Cours pré-requis

Cours co-requis

Méthodes d'enseignement et activités d'apprentissages

 Lectures.
Guided exercises in small groups.

Références, bibliographie et lectures recommandées

D.Barden, C.Thomas. An introduction to differential manifolds, Imperiall College Press.
J.Lee. Introduction to smooth manifolds, Springer Verlag.
L.Tu. An introduction to manifolds, Springer Verlag.
A.Shastri. Elements of differential topology, CRC Press.

Support(s) de cours

  • Syllabus
  • Université virtuelle

Contribution au profil d'enseignement

•    To master the principles of logical reasoning and to be able to base on them a flawless argumentation.
•    To understand how a concept emerges from observations , examples.
•    Understand an abstraction process and its role in the development of a theory.
•    Write a mathematically rigorous solution or a result in a mathematical theory.

Autres renseignements

Informations complémentaires

80 % written exam + 20 % mini-tests during semester

Contacts

andriy.haydys@ulb.be

Campus

Plaine

Evaluation

Méthode(s) d'évaluation

  • Examen écrit
  • Autre

Examen écrit

  • Question ouverte à réponse courte
  • Question ouverte à développement long
  • Question fermée à Choix Multiple (QCM)

Autre

Written exam and mini-tests during semester. 

Construction de la note (en ce compris, la pondération des notes partielles)

The final grade will consist of 80% for the written exam and 20% of mini-tests during semester. 

Langue(s) d'évaluation

  • anglais
  • (éventuellement français )

Programmes