Coursebooks

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:

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
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
Under construction

Lecture
Exercise, TP
Project, other

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