Differential Geometry I, WS 2020/21

Prof. Dr. C. Löh / AG Ammann

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Differential Geometry I

Differential geometry is the study of geometric objects by analytic means. Geometric objects in this context usually are Riemannian manifolds, i.e., smooth manifolds with a Riemannian metric. This allows to define lengths, volumes, angles, ... in the language of (multilinear) analysis. A central concern of differential geometry is to define and compare (intrinsic) notions of curvature of spaces. A particularly fascinating aspect is that many local curvature constraints are reflected in the global shape.

Differential geometry has various applications in the formalisation of Physics, in medical imaging, and also in other fields of theoretical mathematics. For example, certain phenomena in group theory and topology can only be understood via the underlying geometry.

In this course, we will introduce basic notions of differential geometry. This includes, in particular, the Riemannian curvature tensor, sectional curvature, Ricci curvature, and scalar curvature. Moreover, we will study some first global obstructions to curvature constraints.

If all participants agree, this course can be held in German; solutions to the exercises can be handed in in German or English.

Time/Location

Wednesday, 8:30--10:00,
Thursday, 10:15--12:00.

In view of the COVID-19 pandemic, until further notice, this course will be taught remotely, based on: More details:

Exercise class

Read me! Lecture notes, schedules and assignments

The lecture notes will grow during the semester and will be continuously updated. The first upload is scheduled for November 4, after the first lecture. In subsequent weeks with remote teaching, I will always try to provide the material for the whole week n + 1 on the Thursday of week n. Topics covered so far: This course will not follow a single book. Therefore, you should individually compose your own favourite selection of books. A list of suitable books can be found in the lecture notes.

Weekly assignments:

Try me! Quick checks and interactive tools

The quick checks in the lecture notes help to train elementary techniques and terminology. These problems should ideally be easy enough to be solved within a few minutes. Solutions are not to be submitted and will not be graded. These quickchecks will have feedback implemented directly in the pdf.
This feature is based on PDF layers (not on JavaScript) and is supported by many PDF viewers, such as Acrobat Reader, Evince, Foxit Reader, Okular ("new" versions only), .... (Apple Preview seems to have difficulties with this, though.)
Moreover, there will be simple computational exercises on the assignment sheets (which also will not be submitted/graded). These will give the opportunity to practise basic computational techniques.

Ask me! Interactive sessions

There will be a forum and other communication tools linked on the GRIPS page of this course.

Is it for me? Prerequisites



Last change: November 26, 2020.