Topics in Differential Geometry
Faculteit  Science and Engineering 
Jaar  2022/23 
Vakcode  WMMA04005 
Vaknaam  Topics in Differential Geometry 
Niveau(s)  master 
Voertaal  Engels 
Periode  semester I b 
ECTS  5 
Rooster  rooster.rug.nl 
Uitgebreide vaknaam  Topics in Differential Geometry  
Leerdoelen  At the end of the course, the student is able to: 1. reproduce different models of hyperbolic geometry (the upper halfplane, Poincaré disk, hyperboloid, and BeltramiKlein models). Describe their metric, their geodesics and their group of isometries, and the classification of isometries (elliptic, parabolic, hyperbolic). 2. describe how a hyperbolic surface is obtained from the action of a certain Fuchsian group (e.g., the construction of the Bolza surface  the symmetric surface of genus 2), and is able to apply this knowledge to solve standard problems on these topics. 3. describe special geodesics (closed, dense), the systole (shortest closed geodesic) of a hyperbolic surface, the collar theorem and pants decompositions of a hyperbolic surface, and is able to apply this knowledge to solve standard problems on these topics. 4. summarise and give an oral presentation on a scientific paper on hyperbolic geometry (e.g., on minimal pair of pants decomposition or the existence of embedded disks of large radius in closed hyperbolic surfaces). 

Omschrijving  The central theme of this course concerns one of the most important subjects of differential geometry, namely that of surfaces of constant negative curvature (the socalled hyperbolic surfaces). We start with an introduction to twodimensional hyperbolic geometry and its rich history, discuss various models of this geometry, such as the Poincaré disk model and the hyperboloid model in the Minkowski space. Of particular importance are the isometry groups, which for several of these models turn out to be subgroups of the group of Möbius transformations of the extended complex plane. Hyperbolic surfaces are obtained as quotient spaces under the action of certain Fuchsian groups, which are discrete groups of isometries of the hyperbolic plane. In this way hyperbolic surfaces naturally inherit a Riemannian metric of constant negative curvature from the hyperbolic plane. We shall study certain global (metric and topological) features of such surfaces, such as closed geodesics, area, hyperbolic pants decompositions, and the collar theorem. Depending on the interest of the participants, we will also touch upon advanced topics, such as chaos in the geodesic flow on closed hyperbolic surfaces, links between hyperbolic geometry and special relativity, and/or 3dimensional hyperbolic geometry. 

Uren per week  
Onderwijsvorm 
Hoorcollege (LC), Opdracht (ASM), Werkcollege (T)
(Being present at (at least) 75% of the lectures and at the final presentations is compulsory for passing the course. Active participation in tutorials results in a higher final grade according to the grading scheme.) 

Toetsvorm 
Opdracht (AST), Presentatie (P)
(Final grade is computed based on the homework assignments, presentation, and active participation in the tutorials: 0.7 AST + 0.2 P + 0.1 T. In order to pass the course, the average grade for Assignments (AST  three homework sets) should be at least 5. The average grade for Presentation (P) should be at least 5 as well.) 

Vaksoort  master  
Coördinator  dr. N. Martynchuk  
Docent(en)  dr. N. Martynchuk ,prof. dr. G. Vegter  
Verplichte literatuur 


Entreevoorwaarden  The course assumes knowledge of topics treated in the bachelor courses in Mathematics, mainly Multivariable Analysis, Geometry and Dynamical Systems. Familiarity with Manifolds and basic Differential Geometry is a plus, but is not required.  
Opmerkingen  
Opgenomen in 
