Differential geometry II

Covering spaces and properly discontinuous group actions.

Distributions, Fröbenius' theorem and contact structures.

Existence of partitions of unity.

Differential forms and de Rham cohomology.

Integration and Stokes' theorem.

Classification of 2-manifolds via Morse theory (if time permits).
 

To introduce more advanced topics in differential geometry, such as foliations, differential forms, de Rham cohomoloy and prove the general version of Stokes' theorem.

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. 

Cours ayant celui-ci comme co-requis

Theoretical courses (24 hrs) taught in English and homeworks. 

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, Groupes et algèbres de Lie.

References, bibliography, and recommended reading

1. Lee, Jeffrey. Introduction to smooth manifolds. Springer, Graduate texts in mathematics, 2003.
2. Bott, Raoul & Tu, Loring. Differential forms in algebraic topology. Springer, Graduate texts in mathematics, 1982.
3. Hirsch, Morris W. Differential topology. Corrected reprint of the 1976 original. Graduate Texts in Mathematics, 33. Springer-Verlag, New York, 1994.  
4. Lang, Serge. Introduction to differentiable manifolds. Second edition. Universitext. Springer-Verlag, New York, 2002. 
5. 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.
6. Spivak, Michael, A comprehensive introduction to differential geometry. Vol. V. Second edition. Publish or Perish, Inc., Wilmington, Del., 1979.
7. 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. 
8. Donaldson, Simon. Riemann surfaces, Oxford Gradaute Texts in Mathematics, 22. Oxford University Press, Oxford, 2011.

Course notes

• Syllabus
• Université virtuelle

Contacts

Mélanie Bertelson (2.O7.111) - Melanie.Bertelson@ulb.be - 02 650 58 28.

Campus

Plaine

Method(s) of evaluation

• Personal work
• written examination

Personal work

written examination

• Open question with short answer
• Open question with developed answer

Written exam and homeworks which contribute for 1/4th of the final grade.

Mark calculation method (including weighting of intermediary marks)

Final note based on the note for the written exam N_e and the note for the homeworks N_h according to the rule : 3/4 N_e + 1/4 N_h.

Language(s) of evaluation

• english
• (if applicable french )