This entry is from Winter semester 2016/17 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

Students will

• further develop their understanding of curved spaces and sharpen their mathematical intuition in geometric context,
• learn to grasp and describe mathematical properties in a coordinate-free manner,
• relate geometric extremal properties (such as curvature or curve length) to physical principles of variation
• practice mathematical working methods (developing mathematical intuition and its formal justification, training the ability to abstract, reasoning),
• improve their oral communication skills in the exercises by practicing free speech in front of an audience and in 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

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 M.Sc. Computer Science, this module can be attended in the study area Minor subject Mathematics.