Manifolds (MAT510)

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

MAT100 Mathematical Methods 1, MAT110 Linear Algebra, MAT210 Real Analysis, MAT250 Abstract Algebra, MAT300 Vector Analysis, MAT320 Differential Equations

Exam

Course teacher(s)

Course teacher:

Helge Paul Ruddat

Head of Department:

Bjørn Henrik Auestad

Method of work

5-6 hours lecturing and problem solving per week.

Open for

Must meet the admission requirements of one of the study programmes the course is open for.

Course assessment

The faculty decides whether early dialogue will be held in all courses or in selected groups of courses. The aim is to collect student feedback for improvements during the semester. In addition, a digital course evaluation must be conducted at least every three years to gather students’ experiences.

Literature