Postgraduate course

Course description

Objectives and Content


The course develops the theory of smooth manifolds.


One studies smooth manifolds and functions, the tangent/cotangent space, regularity and transversality, constructions on vector bundles, integrability, Riemannian metrics and partition of unity. One develops the theory of flows and locally trivial fibrations.

Learning Outcomes

On completion of the course the student should have the following learning outcomes defined in terms of knowledge, skills and general competence:


The student

  • Knows basic definitions concerning elements of smooth manifolds.
  • Know constructions like the tangent bundle, and the basic theory for this.
  • Know basic theory for regular values and transversality
  • Know basic theory for (pre-)vector bundles and the manipulations of these, as for instance normal bundles and Hom-bundles.
  • Have insight in the theory leading up to the Ehrensmann vibration theorem.


The student

  • Can establish concrete properties of smooth manifolds through calculations and theory.
  • Can construct smooth manifolds
  • Has solid experience and training in reasoning with compounded geometric structures like vector bundles

General competence

The student

  • Has insight in the most important properties of manifolds as they are used in mathematics, physics and modelling.

Semester of Instruction

Required Previous Knowledge
Recommended Previous Knowledge
MAT121 Linear Algebra, MAT212 Functions of Several Variables and MAT242 Topology
Credit Reduction due to Course Overlap
Forms of Assessment
Oral examination
Grading Scale
The grading scale used is A to F. Grade A is the highest passing grade in the grading scale, grade F is a fail.