MATH-322 / 5 crédits

Enseignant: Tsakanikas Nikolaos

Langue: Anglais


Smooth manifolds constitute a certain class of topological spaces which locally look like some Euclidean space R^n and on which one can do calculus. We introduce the key concepts of this subject, such as vector fields, differential forms, etc.


  • topological and smooth manifolds
  • vector bundles
  • tangent space and tangent bundle
  • vector fields, integral curves
  • differential forms, exterior derivative
  • orientation, integration of differential forms
  • Stokes' theorem (and applications)


smooth manifold, tangent space, vector field, differential form, Stokes


Learning Prerequisites

Required courses

Espaces métriques et topologique, Topologie, Analyse III et IV

Learning Outcomes

By the end of the course, the student must be able to:

  • Define and understand the key concepts (differentiable structure, (co)tangent bundle, etc.)
  • Use these concepts to solve problems
  • Prove the main theorems (Stokes, etc.)

Transversal skills

  • Continue to work through difficulties or initial failure to find optimal solutions.
  • Demonstrate a capacity for creativity.
  • Access and evaluate appropriate sources of information.
  • Demonstrate the capacity for critical thinking
  • Assess one's own level of skill acquisition, and plan their on-going learning goals.

Teaching methods

2h lectures + 2h exercises

Expected student activities

  • Attend classes
  • Revise course content
  • Solve exercises
  • Read appropriate literature to understand key concepts

Assessment methods

Written exam.


Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.


Office hours Yes
Assistants Yes

Dans les plans d'études

  • Semestre: Automne
  • Forme de l'examen: Ecrit (session d'hiver)
  • Matière examinée: Differential geometry II - smooth manifolds
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel

Semaine de référence

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