Differential geometry I

•    Smooth manifolds and smooth maps ;
•    Embeddings and Whitney’s theorem ;
•    Tangent and cotangent bundles, vector bundles ;
•    Vector fields and 1-parameter groups of diffeomorphisms ;
•    Indices of smooth maps ;
•    Indices of vector fields and Euler characteristic of manifolds ;
•    Morse functions.
 

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 ;
•    compute the degree of a smooth map between manifolds of equal dimensions ;
•    compute  the index of a vector field and the Euler characteristic of a manifold.

Connaissances et compétences pré-requises ou co-requises

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

Cours pré-requis

Cours co-requis

•    Lectures, including remote lecture on Teams if lecturing in person will not be possible.
•    Weekly question and answer sessions.
•    Guided exercises in small groups.

Références, bibliographie et lectures recommandées

1. 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.

Informations complémentaires

It is intended to provide written lecture notes. 

Contacts

Andriy Haydys, andriy.haydys@ulb.be

Campus

Plaine

Méthode(s) d'évaluation

• Examen écrit
• Travail de groupe

Examen écrit

Travail de groupe

80 % Examen écrit + 20 % Travail de groupe

Langue(s) d'évaluation

• anglais