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German original

Differential Geometry I
(dt. Differentialgeometrie I)

Level, degree of commitment in original study programme Advanced module, compulsory elective module
Forms of teaching and learning,
workload		 Lecture (4 SWS), recitation class (2 SWS),
270 hours (90 h attendance, 180 h private study)
Credit points,
formal requirements		 9 CP
Course requirement: Successful completion of at least 50 percent of the points from the weekly exercises.
Examination type: Written or oral examination
Language,
Grading		 German,
The grading is done with 0 to 15 points according to the examination regulations for study course M.Sc. Mathematics.
Original study programme M.Sc. Mathematik / Vertiefungsbereich Mathematik
Duration,
frequency		 One semester,
Regelmäßig im Wechsel mit anderen intermediate moduleen im Gebiet Analysis/Geometrie
Person in charge of the module's outline Prof. Dr. Ilka Agricola, Prof. Dr. Pablo Ramacher

Contents

  • Surfaces in three-dimensional space, structure equations, first and second fundamental form, Gaussian and mean curvature,
  • Examples of special surfaces (surfaces of revolution, ruled surfaces, minimal surfaces...), fundamental theorem of surface theory
  • Basics of Riemannian geometry: Riemannian manifolds, connections and covariant derivatives, curvature tensor and derived curvature quantities, Einstein spaces, spaces of constant sectional curvature, geodesic curves, geodesic coordinates, exponential map, completeness properties (inner metric, theorem of Hopf-Rinow)
  • physical applications of differential geometry, e.g. in special or general relativity theory

Qualification Goals

The students shall

  • further develop their understanding of curved spaces and sharpen their mathematical intuition in a geometric context,
  • learn to describe and xpress mathematical properties in a coordinate free way,
  • learn to associate extremal geometric properties (such as curvature or curve length) with physical variation principles,
  • practice mathematical working methods (development of mathematical intuition and its formal justification, training of the ability to abstract, proof techniques),
  • improve their oral communication skills in the exercises by practicing free speech in front of an audience and during discussion.

Prerequisites

None. The competences taught in the following modules are recommended: either Linear Algebra I and Linear Algebra II or Basic Linear Algebra, either Analysis I and Analysis II or Basic Real Analysis, Complex Analysis and Vector Analysis.


Recommended Reading

  • Barret O'Neill, Semi-Riemannian geometry. Academic Press, 1983.
  • Michael Spivak, A comprehensive introduction to differential geometry, Berkeley, California: Publish Perish, Inc.



Please note:

This page describes a module according to the latest valid module guide in Wintersemester 2020/21. Most rules valid for a module are not covered by the examination regulations and can therefore be updated on a semesterly basis. The following versions are available in the online module guide:

The module guide contains all modules, independent of the current event offer. Please compare the current course catalogue in Marvin.

The information in this online module guide was created automatically. Legally binding is only the information in the examination regulations (Prüfungsordnung). If you notice any discrepancies or errors, we would be grateful for any advice.