Differential Geometry (Differentialgeometrie)

Lecture Winter Term 2016/2017

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Preliminary Program

This lecture aims at students in the programs International Master in Mathematics, in Mathematics, and Mathematical Physics. It may also be interesting for students in the master program Physics.

If there are students of the International Master in Mathematics then the lecture can of course be offered in English. Otherwise the lecture will be in German. The lecture notes will be in English anyway. Please get in contact with me early enough then we can discuss and arrange the details.

This master course is a first introduction to the topics of differential geometry. We will discuss differentiable manifolds as geometric objects in an intrinsic approach. A particular emphasize will be put on the global calculus on manifolds, showing how the coordinate-based calculations can be minimized as far as possible. After manifolds, we discuss vector bundles as the next important ingredient in differential geometry. Integration on manifolds will be presented in two ways, using an orientation and without orientation. We will see the most important cohomology theories attached to manifolds. If time permits, we will also give a short introduction to Lie groups.

The lecture will have a second part in the following summer term: this will be most probably an introduction to geometric mechanics, symplectic and Poisson manifolds. Beside that, this lecture serves as the starting point of various other courses in the above master program: there will be seminars and RiGs where are a good understanding of differential geometry as provided in this course is mandatory. For students of mathematical physics, this lecture will be (depending on the lecturer) also a starting point for the Analysis und Geometrie klassischer Systeme. Of course, it can also serve as background to lectures in general relativity.

  1. Differentiable manifolds
  2. Vector bundles and their sections
  3. Calculus on manifolds
  4. Integration and cohomology
  5. Lie-Groups, Lie algebras, and their actions


We expect good knowledge from the bachelor courses in analysis and linear algebra. In particular, we will need some aspects of multilinear algebra and tensor products (which will be briefly recalled if necessary). The bachelor course Geometrische Analysis can be seen as a motivation: there one considers submanifolds of the euclidean space, now we treat manifolds intrinsically. However, this course will not be required. Finally, some basic knowledge in point-set topology will be useful: we will recall the relevant information if necessary.


The list of textbooks is rather long, there are many good texts in differential geometry. Many of the references should be seen as background information. At the beginning of the lecture we will point out some more details to particular texts.



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