This entry is from Summer semester 2018 and might be obsolete. You can find a current equivalent here.

# 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): Written or oral examination Examination type: Successful completion of at least 50 percent of the points from the weekly exercises. Language,Grading German,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. 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

Translation is missing. Here is the German original:

Keine. Empfohlen werden die Kompetenzen, die in den Basismodulen sowie im Aufbaumodul Funktionentheorie und Vektoranalysis vermittelt werden.

## Applicability

The module can be attended at FB12 in study program(s)

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

When studying M.Sc. Mathematics, this module can be attended in the study area Specialization Modules in Mathematics.

The module can also be used in other study programs (export module).

Die Wahlmöglichkeit des Moduls ist dadurch beschränkt, dass es der Reinen Mathematics zugeordnet ist.