Exercise session of differential geometry 1 [diffgeo]

This page is reserved for the course “Grundlagen der Geometrie und Topologie” taught in SS2025 in Heidelberg. I will upload my notes for the exercise sessions here. All these notes are not official and written by my own. They should only be considered as the supplementary material for the preparation of exams.

1. Week 0: Point-set topology [diffgeo-w0]

In the 0th week we have reviewed some concepts of basic point-set topology, especially the notion of product topology and compactness.

2. Week 1: Topological and smooth manifolds [diffgeo-w1]

In the 1st week we have seen more examples of manifolds, talked about the orientability, filled the gap of the rank theorem in the lecture and studied the smooth group actions on manifolds.

3. Week 2 [diffgeo-w2]

    Cancelled due to public holidays.

    4. Week 3: Tangent spaces and vector bundles [diffgeo-w3]

    This week is mainly about an important construction in differential geometry: the vector bundles. A special case is the tangent bundle of a smooth manifold, and we explained some geometric intuition of this object.

    5. Week 4: Metrizable manifolds and distributions [diffgeo-w4]

    We give a geometric intuition of how distributions look like and explore the two theorems of Frobenius. In the second part we prove that every manifold is metrizable, which is a very useful fact.

    6. Week 5: Partition of unity [diffgeo-w5]

    We prove the Tietze extension theorem and existence of partition of unity on manifolds, both using the beautiful Urysohn’s lemma. Partition of unity is indeed a highlight of differential geometry, to make many things on manifolds possible.

    7. Week 6 + 7: Lie groups and Lie algebras [diffgeo-w6w7]

    [In preparation] We cover some important examples of Lie groups and Lie algebras, together with some themes that is not covered in the lecture, including Lie group representations and Baker-Hausdorff formula.