# Fiches de cours 2018-2019

## Introduction to differentiable manifolds

Kiesenhofer Anna

English

#### Summary

Differentiable manifolds are (certain) topological spaces which, in a way we will make precise, locally resemble R^n. We introduce the key concepts of this subject, such as vector fields, differential forms, integration of differential forms etc.

#### Content

• topological and differentiable manifolds, partitions of unity
• vector bundles
• tangent space and tangent bundle
• vector fields, integral curves
• differential forms, tensors, exterior derivative
• orientation, integration of differential forms
• Stokes's theorem (and applications)
• if time permits further topics such as de Rham cohomology, Riemannian manifolds, geodesics,...

#### Keywords

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

#### Learning Prerequisites

##### Required courses

Espaces métriques et topologique, Topologie, Analyse III et IV

##### Important concepts to start the course

Topological spaces, multivariate calculus (implicit function theorem etc.)

#### Learning Outcomes

By the end of the course, the student must be able to:
• Define and unerstand 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 and solve exercises, revise course content / read appropriate literature to understand key concepts.

Written exam.

#### Resources

##### Bibliography

John M. Lee: Introduction to Smooth Manifolds (e-book:

### Dans les plans d'études

• Semestre
Automne
• Forme de l'examen
Ecrit
• Crédits
5
• Matière examinée
Introduction to differentiable manifolds
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
2 Heure(s) hebdo x 14 semaines

### Semaine de référence

LuMaMeJeVe
8-9 MAA330
9-10
10-11 MAA330
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22

Cours
Exercice, TP
Projet, autre

### légende

• Semestre d'automne
• Session d'hiver
• Semestre de printemps
• Session d'été
• Cours en français
• Cours en anglais
• Cours en allemand