Differential geometry II - smooth manifolds

Summary

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.

Content

Keywords

smooth manifold, tangent space, vector fields, differential forms, Stokes' theorem

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 regularly
• 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.

Supervision

Resources

Bibliography

John M. Lee: Introduction to Smooth Manifolds

Jefrrey M. Lee: Manifolds and Differential Geometry

Ressources en bibliothèque

• Introduction to Smooth Manifolds / Lee
• Manifolds and Differential Geometry / Lee

Websites

• https://slsp-epfl.primo.exlibrisgroup.com/discovery/search?tab=41SLSP_EPF_MyInst_and_CI&search_scope=MyInst_and_CI&vid=41SLSP_EPF:prod&facet=rtype,include,books&query=isbn,contains,978-1-4419-9981-8
• https://link.springer.com/book/10.1007/978-1-4419-9982-5
• https://slsp-epfl.primo.exlibrisgroup.com/discovery/fulldisplay?docid=cdi_proquest_ebookcentral_EBC3114665&context=PC&vid=41SLSP_EPF:prod&lang=fr&search_scope=MyInst_and_CI&adaptor=Primo%20Central&tab=41SLSP_EPF_MyInst_and_CI&query=any,contains,Jeffrey%20m%20lee%20manifolds&offset=0
• https://www.ams.org/books/gsm/107/

Moodle Link

• https://go.epfl.ch/MATH-322

Prerequisite for

Differential Geometry III - Riemannian Geometry

Differential Geometry IV - General Relativity