# Coursebooks

## Introduction to differentiable manifolds

Lodha Yash

English

#### Summary

Differentiable manifolds are a certain class of 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
• 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)

#### 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 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 and solve exercises, revise course content / 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

#### Resources

##### Bibliography

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

### In the programs

• Semester
Fall
• Exam form
Written
• Credits
5
• Subject examined
Introduction to differentiable manifolds
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks

### Reference week

MoTuWeThFr
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

Lecture
Exercise, TP
Project, other

### legend

• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German