Differential Geometry I/Differentialgeometrie I

Winter term 2023/24

Prof. Bernd Ammann, Office M 119

Current Announcements

Please register on GRIPS. We will send email announcements over this system.

Language/Sprache

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 English.

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.

Literature

Scripts

C. Bär, Differential Geometry (Unpublished Lecture Notes), Summer Term 2023, (German version from a previous lecture; Old version from Summer Term 2013),
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)

Books

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

Conventions

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.
A topological manifold is defined as a locally Euclidean Hausdorff space with a countable basis of topology.

Exercise groups

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

Exercise Sheets

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

Examination/Prüfung

The examination is an oral examination of about 30 minutes.
we will offer several alternatives and you also ma suggest other dates.
The examination can take place during the week Feb 5th to Feb 9th (last week of the teaching period), or after February 26th.

Formal criteria/Kriterien für Leistungsnachweise

We refer to the KVV.

Event number (Veranstaltungsnummer, z.B. in SPUR)

51106: Vorlesung
51107: Übungen

Bernd Ammann, 31.01.2024
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