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 and integral curves 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 and Complex Calculus, MAT250 Abstract Algebra, MAT300 Vector Analysis, MAT320 Differential Equations

Exam

Course teacher(s)

Course coordinator:

Paul Francis de Medeiros

Head of Department:

Bjørn Henrik Auestad

Method of work

5-6 hours lecturing and problem solving per week.

Open for

Course assessment

There must be an early dialogue between the course coordinator, the student 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.

Literature