Manifolds

In this course, we will discuss the application of differential forms to the geometry of manifolds. Manifolds are "smooth" geometric objects, such as spheres and tori. Differential forms are a generalization of functions. Just as a function can be evaluated at a point, so can a differential form of degree d be integrated over a d-dimensional submanifold to yield a number.

Using differential forms one can construct invariants of manifolds, which give for example a classification of all compact orientable surfaces.   

Key words:

* Differential forms on Rⁿ
* Differentiable manifolds 
* Integration of Differential forms (Stokes' theorem)
* Poincare Lemma, Mayer-Vietoris, und Poincare Duality
* Euler charakteristic and the Lefschetz Fix point theorem
* Characteristic classes of vector bundles

Texts

Main Text:
Bott, Raoul; Tu, Loring W. Differential forms in algebraic topology. Graduate Texts in Mathematics, 82. Springer-Verlag, New York-Berlin, 1982.

Supplemental texts:
Jänich - Vektoranalysis, Springer, 2000
Madsen, Tornehave - From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes, Cambridge University Press, 1997

Schedule

The lectures will be held Mondays 12-14h and Fridays 10-12h in Weststadt Carree / WSC-S-U-3.03.