Differential Geometry I/Differentialgeometrie I
Winter term 2023/24
Please register on GRIPS. We will send email announcements over this system.
This course is "English on demand". This means the language will be decided
during the first week of the lecture. If there is at least one
non-German-speaking person in the audience
who is sincerely following the lecture, then the lecture will be given in
Time and Location
Monday and Friday 10-12, in M102.
Recommended previous knowledge
- Analysis I, II and IV
- Linear Algebra I and II
Content of the Lecture
This lecture is an introduction to differential geometry, more precisely to semi-Riemannian manifolds, their curvature and global properties.
The main topic are Riemannian metrics on manifolds. The simplest examples are surfaces in Euclidean space ℝ3. Such surfaces may be intrinsically curved, as e.g. the sphere. Or they may only be extrinsically curved, as e.g. a cylinder -- which may be cut by a "scissor" and then this surface is isometric to an open set of a plane.
The goal is to understand not only surfaces, but similar curvature quantities in arbitrary dimensions and codimensions, a generalization going back to work of Bernhard Riemann. Very similar structures were later used by Einstein and others in order to get a mathematical framework to describe general relativity.
The theory is still a very active area in mathematics and theoretical physics.
The lecture will be continued in the summer term.
- C. Bär, Differential Geometry (Unpublished Lecture Notes), Summer Term 2023,
(German version from a previous lecture; Old version from Summer Term 2013),
(German version from a previous lecture)
- B. Ammann,
Lineare Algebra I, WS 2007/08
- B. Ammann,
Analysis I+II, 2018/19
- B. Ammann,
Analysis III, WS 2019/20
- B. Ammann,
Analysis IV, SS 2020
- B. Ammann, Kurzskript zu Tensoren (German language)
- N. Ginoux (former member of my group), Kurzskript zur Topologie (German language)
- M. do Carmo, Riemannian Geometry, Birkhäuser
- Cheeger, Ebin, Comparison theorems in Riemannian Geometry
- B. O'Neill, Semi-Riemannian geometry. With applications to relativity.
Pure and Applied Mathematics, 103. Academic Press
- F. Warner, Foundations of differentiable manifolds and Lie groups, Springer
- T. Sakai, Riemannian Geometry, Transl. Math. Monogr., AMS
- W. Kühnel, Differentialgeometrie, Vieweg
- J. Lee, Introduction to topological manifolds, Springer
- J. Lee, Introduction to smooth manifolds, Springer
- J. Lee, Riemannian manifolds, Springer
- Zero is not a natural number in this lecture
- The empty set is connected and path-connected
- A topological space is quasi-compact, if and only if each open covering admits a finite subcovering. Compact spaces are defined as quasi-compact Hausdorff spaces.
- We use the term "paracompact"
as explained on this Wikipedia page. However, it will be rarely used.
A topological manifold is defined as a locally euclidean Hausdorff space, whose connected components have a countable basis of topology. There is no restriction on the number of connected components, we also allow uncountably many. A smooth manifold is a topological manifold with a smooth atlas. This is equivalent to having a paracompact locally euclidean Hausdorff space with a smooth atlas.
We recommend that any students takes part in one of the exercise groups.
Currently two exercise groups are planned:
- Thursday 14-16, M009
- Friday 12-14, M102
There will be a weekly exercise sheet, the solutions will be presented and discussed in the exercise groups.
and finally all exercise sheets in one pdf file.
Links to related web pages
The examination is an oral examination of about 30 minutes.
Formal criteria/Kriterien für Leistungsnachweise
We refer to the KVV.
Event number (Veranstaltungsnummer, z.B. in SPUR)
Bernd Ammann, 27.09.2023
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