Schedule, syllabus and examination date

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Exams after the reopening

As a general rule, exams will be conducted without physical attendance in the autumn of 2021, even after the reopening. See the semester page for information about the form of examination in your course. See also more information about examination at the MN Faculty in 2021.

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 completing the course you:

  • understand well the concepts smooth manifold, smoth map, and tangent space;
  • know how the inverse function theorem can be used to describe the local structure of immersions and submersions, and you know Sard´s theorem;
  • can work with submanifolds and know Whitney´s embedding theorem;
  • 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;
  • are familiar with the Lie algebra and exponential map of a Lie group;
  • 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;
  • will present, on a scientific level, a short thesis on a chosen topic of relevance, selected in collaboration with the lecturer.

Admission to the course

PhD candidates from the Faculty of Mathematics and Natural Sciences at 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.

MAT3500 – Topology/MAT4500 – Topology and MAT4510 – Geometric Structures.

Overlapping courses

  • 10 credits overlap with MAT4520 – Manifolds.
  • 10 credits overlap with MA152.
  • 10 credits overlap with MA252.
  • 10 credits overlap with MA352.
  • 10 credits overlap with MA352.


4 hours of lectures/exercises per week.

The course may be taught in Norwegian if the lecturer and all students at the first lecture agree to it.

Upon the attendance of three or fewer students, the lecturer may, in conjunction with the Head of Teaching, change the course to self-study with supervision.


Final oral exam which counts 100 % towards the final grade.

In addition, each PhD candidate is expected to give an oral presentation on a topic of relevance chosen in cooperation with the lecturer. The presentation has to be approved by the lecturer before you can sit the final exam.

It will also be counted as one of the three attempts to sit the exam for this course, if you sit the exam for one of the following courses: MAT4520 – Manifolds

Examination support material

No examination support material is allowed.

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.

Resit an examination

This course offers both postponed and resit of examination. Read more:

Special examination arrangements, use of sources, explanations and appeals

See more about examinations at UiO

Last updated from FS (Common Student System) Dec. 2, 2021 9:36:43 AM

Facts about this course

Teaching language