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Differential Geometry I
(dt. Differentialgeometrie I)

Level, degree of commitment Specialization 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(s): Successful completion of at least 50 percent of the points from the weekly exercises.
Examination type: Written or oral examination (individual examination)
Language,
Grading		 English,
The grading is done with 0 to 15 points according to the examination regulations for the degree program M.Sc. Mathematics.
Duration,
frequency		 One semester,
Regularly alternating with other advanced modules im Gebiet Analysis/Geometrie
Person in charge of the module's outline Prof. Dr. Ilka Agricola, Prof. Dr. Oliver Goertsches, Prof. Dr. Pablo Ramacher

Contents

  • Surfaces in three-dimensional space, structural equations, first and second fundamental forms, Gaussian and mean curvature,
  • Examples of special surfaces (surfaces of revolution, ruled surfaces, minimal surfaces...), fundamental theorem of surface theory.
  • Fundamentals of Riemannian geometry: Riemannian manifolds, relations and covariant derivatives, curvature tensor and derived curvature quantities, Einstein spaces, spaces of constant sectional curvature, geodesic curves, geodesic coordinates, exponential mapping, completeness properties (inner metric, theorem of Hopf-Rinow), theorems of global Riemannian geometry (Gauss-Bonnet, Bonnet-Myers, Synge)
  • Physical applications of differential geometry, for example in special or general relativity theory

Qualification Goals

Students will

  • possess an understanding of curved spaces and have sharpened their mathematical intuition in geometric context,
  • can grasp and describe mathematical properties in a coordinate-free manner,
  • can relate geometric extremal properties (such as curvature or curve length) to physical principles of variation,
  • have deepened mathematical working methods (developing mathematical intuition and its formal justification, abstraction, proof),
  • have improved their oral communication skills in exercises by practicing free speech in front of an audience and in 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.


Applicability

Module imported from M.Sc. Mathematics.

It can be attended at FB12 in study program(s)

  • B.Sc. Mathematics
  • M.Sc. Computer Science
  • M.Sc. Mathematics
  • LAaG Mathematics

When studying B.Sc. Mathematics, this module can be attended in the study area Compulsory Elective Modules in Mathematics.


Recommended Reading

  • Manfredo Perdigão do Carmo, Riemannian geometry, Birkhäuser.
  • Detlef Gromoll, Wilhelm Klingenberg, Wolfgang Meyer, Riemannsche Geometrie im Großen, Springer.
  • Barret O'Neill, Semi-Riemannian geometry, Academic Press.
  • 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 Winter semester 2023/24. 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.