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
- topological and smooth manifolds
- tangent space and tangent bundle
- vector fields, integral curves
- cotangent space and cotangent bundle
- differential forms, exterior derivative
- orientations of manifolds
- Integration of differential forms and Stokes' theorem
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
| Office hours | No |
| Assistants | Yes |
| Forum | Yes |
Resources
Bibliography
John M. Lee: Introduction to Smooth Manifolds
Jefrrey M. Lee: Manifolds and Differential Geometry
Ressources en bibliothèque
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
Prerequisite for
Differential Geometry III - Riemannian Geometry
Differential Geometry IV - General Relativity
In the programs
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Differential geometry II - smooth manifolds
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional