# 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

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 Compulsory Elective Modules in Mathematics.

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

The module is assigned to Pure Mathematics. Further information on eligibility can be found in the description of the study area.

• 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.