Differential Geometry

Module MA3205

This Module is offered by TUM Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

Module version of SS 2021 (current)

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions
SS 2021	SS 2020	SS 2012	WS 2011/2

Basic Information

MA3205 is a semester module in English language at Master’s level which is offered in winter semester.

This Module is included in the following catalogues within the study programs in physics.

Catalogue of non-physics elective courses
Specialization Modules in Elite-Master Program Theoretical and Mathematical Physics (TMP)

Total workload	Contact hours	Credits (ECTS)
270 h	90 h	9 CP

Content, Learning Outcome and Preconditions

Content

Smooth Manifolds, tangential, vector, and principal bundles, Riemannian metrics, curvature tensor, geodesics, symmetric spaces, Liegroups.

Learning Outcome

After successful completion of this course, the students know how metric properties of the euclidean space are extended to "curved" spaces. They are able to apply this curvature theory to derive properties of these spaces using analytic methods.

Preconditions

MA1001 Analysis 1, MA1002 Analysis 2, MA1101 Linear Algebra and Discrete Structures 1, MA1102 Linear Algebra and Discrete Structures 2, MA2204 Elementary Differential Geometry

Courses, Learning and Teaching Methods and Literature

Courses and Schedule

Please keep in mind that course announcements are regularly only completed in the semester before.

WS 2020/1 WS 2019/20 WS 2018/9 WS 2016/7 WS 2014/5 WS 2012/3

Type	SWS	Title	Lecturer(s)	Dates	Links
VO	1	Advanced Topics in Differential Geometry	Hoffmann, T.	Thu, 10:00–12:00, MI 02.06.011
VO	4	Differential Geometry	Hoffmann, T.	Tue, 10:15–11:45, MI HS3 Thu, 16:00–18:00, PH HS1	eLearning
UE	2	Exercises for Differential Geometry	Hoffmann, T. Ye, Z.	dates in groups

Learning and Teaching Methods

lecture, exercise course, self-study assignments
The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should motivate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress.

Media

blackboard or computer

Literature

do Carmo: Riemannian Geometry, Birkhäuser, 1992. Helgason: Differential Geometry, Lie Groups, and Symmetric Spaces, AMS 2001.
Kühnel: Differential Geometry: Curves - Surfaces - Manifolds, AMS, 2005.

Module Exam

Description of exams and course work

The module examination is based on a written exam (60-90 minutes). Students show their ability to extend the metric properties of Euclidean spaces to curved spaces and can apply this theory to independently develop properties of these spaces with analytic methods in a comprehensible way.

Exam Repetition

The exam may be repeated at the end of the semester.