Course
Manifolds (MAT510)
An introduction to smooth manifolds and related concepts in differential geometry.
Course description for study year 2025-2026. Please note that changes may occur.
Facts
Course code
MAT510
Credits (ECTS)
10
Semester tution start
Autumn
Language of instruction
English
Number of semesters
1
Exam semester
Autumn
Time table
Literature
Content
This course gives an introduction to smooth manifolds and related concepts in differential geometry. A brief review of essential preliminaries will be provided, including fundamental elementary concepts like sets, maps, groups and algebras. The basics of point-set topology will be covered, followed by a presentation of smooth maps, directional derivatives and tangent vectors in Euclidean space that will be apt to generalise to smooth manifolds. The notion of a smooth manifold will be introduced, with a plethora of familiar (and perhaps not so familiar) examples. Many important related concepts like smooth maps, diffeomorphisms, tangent spaces, differentials, smooth curves, submanifolds, vector fields, integral curves, Lie groups and Lie algebras will also be developed.
Learning outcome
After completing this course, the student should understand how familiar concepts from differential calculus in Euclidean space are subsumed by the framework of smooth manifolds. In particular, the student should be able to state key definitions, perform simple calculations on smooth manifolds and work out detailed properties in examples.
Required prerequisite knowledge
None
Recommended prerequisites
Mathematical Methods 1 (MAT100), Linear Algebra (MAT110), Real and Complex Calculus (MAT210), Abstract Algebra (MAT250), Vector Analysis (MAT300), Differential Equations (MAT320)
Exam
Course teacher(s)
Head of Department:
Bjørn Henrik AuestadCourse coordinator:
Paul Francis de MedeirosMethod of work
5-6 hours lecturing and problem solving per week.
Overlapping courses
| Course | Reduction (SP) |
|---|---|
| Mathematical Modelling (MAT500_1) , Manifolds (MAT510_1) | 10 |
Open for
Admission to Single Courses at the Faculty of Science and Technology
Mathematics and Physics - Master of Science Degree Programme
Mathematics and Physics - Five Year Integrated Master's Degree Programme
Course assessment
There must be an early dialogue between the course supervisor, the student union representative and the students. The purpose is feedback from the students for changes and adjustments in the course for the current semester.In addition, a digital course evaluation must be carried out at least every three years. Its purpose is to gather the students experiences with the course.
The course description is retrieved from FS (Felles studentsystem). Version 1