Course teacher(s)
Mélanie BERTELSON-VOLCKAERT (Coordinator)ECTS credits
5
Language(s) of instruction
english
Course content
Review of the basic concepts
Covering spaces and properly discontinuous group actions.
Distributions, Fröbenius' theorem foliations and contact structures.
Differential forms and de Rham cohomology.
Integration and Stokes' theorem.
Objectives (and/or specific learning outcomes)
To introduce more advanced topics in differential geometry, such as foliations, differential forms, de Rham cohomoloy and prove the general version of Stokes' theorem.
Prerequisites and Corequisites
Required and Corequired knowledge and skills
To be able to follow this course, It is necessary to have passed a first course on differential geometry, such as MathF310, covering at least the notion of :
- intrinsic manifold
- smooth map between smooth manifolds
- tangent vector and tangent space at a point in a manifold
- vector field and its flow
- differential of a smooth map
- immersions, submersions and their local normal forms
- Immersed and embedded submanifolds.
Cours ayant celui-ci comme co-requis
Teaching methods and learning activities
Theoretical courses (24 hrs), exercise sessions (24 hrs), both taught in English, as well as homeworks.
References, bibliography, and recommended reading
- Lee, Jeffrey. Introduction to smooth manifolds. Springer, Graduate texts in mathematics, 2003.
- Bott, Raoul & Tu, Loring. Differential forms in algebraic topology. Springer, Graduate texts in mathematics, 1982.
- Hirsch, Morris W. Differential topology. Corrected reprint of the 1976 original. Graduate Texts in Mathematics, 33. Springer-Verlag, New York, 1994.
- Lang, Serge. Introduction to differentiable manifolds. Second edition. Universitext. Springer-Verlag, New York, 2002.
- Milnor, John. Morse theory. Based on lecture notes by M. Spivak and R.Wells. Annals ofMathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963.
- Spivak, Michael, A comprehensive introduction to differential geometry. Vol. V. Second edition. Publish or Perish, Inc., Wilmington, Del., 1979.
- Warner, Frank W.Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983.
- Donaldson, Simon. Riemann surfaces, Oxford Gradaute Texts in Mathematics, 22. Oxford University Press, Oxford, 2011.
Course notes
- Syllabus
- Université virtuelle
Contribution to the teaching profile
This course is about manifolds without additional structures and follows MathF310. It is co-requis of the following courses : Géométrie Riemannienne, Géométrie Symplectique, Riemann Surfaces, Global Analysis.
Other information
Additional information
The use of AI, whether to study or to solve the exercise sheets or the problems, is strongly discouraged.
Contacts
Mélanie Bertelson (2.O7.111) - Melanie.Bertelson@ulb.be - 02 650 58 28.
Friso van Dijk (2.O7.213) - friso.van.dijk@ulb.be.
Campus
Plaine
Evaluation
Method(s) of evaluation
- Personal work
- written examination
Personal work
written examination
- Open question with short answer
- Open question with developed answer
A 3-hour written examination consisting of both theoretical questions and exercises, together with assignments involving problem sets given every three weeks. The questionnaires will be written in English, but answers may be written in French.
Mark calculation method (including weighting of intermediary marks)
The examination will be graded out of 15 points, denoted Ne.
The assignments will be graded out of 5 points, denoted Nd.
The final grade is the sum of these two grades, provided that the examination is passed, that is, provided that the examination grade is greater than 7.5/15. Otherwise, only the examination grade is taken into account; in other words, the final grade is equal to 4/3N_e.
Language(s) of evaluation
- english
- (if applicable french )