Schedule, syllabus and examination date

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Course content

The course gives an introduction to smooth manifolds. It covers tangent bundles, vector fields and integral curves (ordinary differential equations), Lie groups, differential forms, and integration. This theory is fundamental to both modern geometry and theoretical physics.

Learning outcome

After having completed the course

• you understand well the concepts smooth manifold, smoth map, and tangent space

• you know how the inverse function theorem can be used to describe the local structure of immersions and submersions, and you know Sard's theorem

• you can work with submanifolds and know Whitney's embedding theorem

• you know fundamental results about vector fields, Lie brackets, and integral curves, and you know what it means for the flows of two vector fields to commute

• you are familiar with the Lie algebra and exponential map of a Lie group

• you can do calculations with differential forms and characterise the exterior derivative, and you know Stokes' theorem and understand how this generalises classical

theorems in calculus

After having completed the course you will also be able to:

• present, on a scientific level, a short thesis on a chosen topic of relevance, selected in collaboration with the lecturer


PhD candidates from the University of Oslo should apply for classes and register for examinations through Studentweb.

If a course has limited intake capacity, priority will be given to PhD candidates who follow an individual education plan where this particular course is included. Some national researchers’ schools may have specific rules for ranking applicants for courses with limited intake capacity.

PhD candidates who have been admitted to another higher education institution must apply for a position as a visiting student within a given deadline.


Formal prerequisite knowledge


Recommended previous knowledge

MAT3500 - Topology/MAT4500 - Topology and MAT4510 - Geometric structures.

Overlapping courses

10 credits overlap with MA152

10 credits with MA252/352.

10 credits with MAT4520 - Manifolds.

*The information about the overlaps is not complete. Contact the Department for more information if necessary.


4 hours of lectures/exercises per week.


Oral exam.

In addition, each phd student is expected to give a one hour oral presentation on a topic of relevance (chosen in cooperation with the lecturer). The presentation has to be approved by the lecturer for the student to be admitted to the final exam.

Language of examination

Subjects taught in English will only offer the exam paper in English.

You may write your examination paper in Norwegian, Swedish, Danish or English.

Grading scale

Grades are awarded on a pass/fail scale. Read more about the grading system.

Explanations and appeals

Resit an examination

Students who can document a valid reason for absence from the regular examination are offered a postponed examination at the beginning of the next semester.

Re-scheduled examinations are not offered to students who withdraw during, or did not pass the original examination.

Withdrawal from an examination

It is possible to take the exam up to 3 times. If you withdraw from the exam after the deadline or during the exam, this will be counted as an examination attempt.

Special examination arrangements

Application form, deadline and requirements for special examination arrangements.


The course is subject to continuous evaluation. At regular intervals we also ask students to participate in a more comprehensive evaluation.

Facts about this course






Every spring


Every spring

Teaching language


The course is given in English. If no students have asked for the course in English within the first lecture, it may be given in Norwegian.