Caput Differential Geometry
Faculteit  Science and Engineering 
Jaar  2020/21 
Vakcode  WMMA01005 
Vaknaam  Caput Differential Geometry 
Niveau(s)  master 
Voertaal  Engels 
Periode  semester I b 
ECTS  5 
Rooster  rooster.rug.nl 
Uitgebreide vaknaam  Caput Differential Geometry  
Leerdoelen  At the end of the course the student is able to: 1. reproduce two conformal models of the hyperbolic plane, and describe their metric, their geodesics and their group of isometries, the classification of isometries (elliptic, parabolic, hyperbolic), and is able to apply this knowledge to solve standard problems on these topics. 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 a scientific paper on hyperbolic surfaces (e.g., on shortest geodesics on compact hyperbolic surfaces, or on the construction of a complete hyperbolic metric on certain surfaces) by writing a review paper. 5. give an oral presentation of the written review of the assigned scientific paper. 

Omschrijving  The central theme of this course is the differential geometry of surfaces of constant curvature, like the sphere (positive curvature), flat tori (zero curvature) and hyperbolic surfaces (negative curvature). The main focus is on hyperbolic surfaces. We start with an introduction to twodimensional hyperbolic geometry and its history, discuss various models of this geometry, and show how they can be endowed with a metric of constant negative curvature. Of particular importance are the isometry groups, which for several of these models turn out to be subgroups of the group of Mobius transformations. 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. Metric features like lengths of curves, geodesics (locally shortest curves) and area are of particular interest in the classification of hyperbolic surfaces up to isometry. 

Uren per week  
Onderwijsvorm  Bijeenkomst (S), Hoorcollege (LC), Opdracht (ASM)  
Toetsvorm 
Opdracht (AST), Presentatie (P), Verslag (R)
(Final grade = 0.25 R + 0.25 P + 0.5 AST. Average grade for Report 1 (R) and Presentation 1 (P) should be at least 5. Average grade for Assignment 1 (AST  three homework sets) should be at least 5. Be present at (at least) 75% of the (12) lectures and at 100% of the presentations. Missing out means failing the course.) 

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


Entreevoorwaarden  Knowledge of topics treated in the bachelor courses Analysis on Manifolds and (Differential) Geometry.  
Opmerkingen  This course was registered last year with course code WMMA14001 

Opgenomen in 
