Caput Differential Geometry

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Faculteit Science and Engineering
Jaar 2022/23
Vakcode WMMA010-05
Vaknaam Caput Differential Geometry
Niveau(s) master
Voertaal Engels
Periode semester I b
ECTS 5
Rooster Be present at (at least) 75% of the (12) lectures and at the presentations. Missing out (without a good reason) means failing the course.

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 two-dimensional 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 = 0.1 R + 0.2 P + 0.7 AST Average grade for Report 1 (R) and Presentation 1 (P) should be at least 5. Average grade for Assignment (AST - three homework sets) should be at least 5.)
Vaksoort master
Coördinator dr. N. Martynchuk
Docent(en) dr. N. Martynchuk ,prof. dr. G. Vegter
Verplichte literatuur
Titel Auteur ISBN Prijs
Geometry of Surfaces, 2nd edition, 2012 J. Stillwell 1461209293, 97814612 €  53,50
Hyperbolic Geometry, 2nd edition, 2008 J. Anderson 1852339349 €  37,40
The geometry of discrete groups, 2nd edition, 1995 A.F Beardon 0387907882 €  63,00
Hyperbolic Geometry (lecture notes) C. Series
Slides of Lectures and Lecture notes G. Vegter
Entreevoorwaarden Knowledge of topics treated in the bachelor courses, including Geometry and Analysis on Manifolds
Opmerkingen
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