Geometry
Faculteit  Science and Engineering 
Jaar  2022/23 
Vakcode  WBMA03405 
Vaknaam  Geometry 
Niveau(s)  bachelor 
Voertaal  Engels 
Periode  semester II a 
ECTS  5 
Rooster  rooster.rug.nl 
Uitgebreide vaknaam  Geometry  
Leerdoelen  At the end of the course, the student is able to: 1. define and explain the differences between the classical geometries treated in this course: affine, Euclidean, projective, hyperbolic, and spherical geometry. Describe the corresponding isometry groups. 2. chose an appropriate geometry to solve specific problems, using tools from linear algebra and multivariable analysis. 3. explain what is a Riemannian metric of a 2dimensional surface and how it gives rise to a geometry on this surface. This includes describing geodesics and other geometric quantities, such as lengths and volumes, in standard examples. 4. define the curvature and torsions of a space curve and compute these invariants in specific examples. Show how the notion of curvature (in the case of plane curves) generalises to affine and projective geometries. 5. define and compute Gaussian curvature (for specific Riemannian metrics on 2surfaces) and explain its relation to Euclidean, hyperbolic, and spherical geometry. 

Omschrijving  Geometry has its root in what we now call Euclidean geometry of idealised figures (such as points, lines, and surfaces), their mutual distances and relative positions, as summarised in Euclid's Elements or, more formally, using Hilbert's axioms. Euclidean geometry had been a dominating theory of geometry until the 19th century when it was realised that some of Euclid's postulates can be considered independently of the rest or even replaced by other postulates (the cornerstone being the 5th parallel postulate). This prompted the development of a variety of different geometries, which are extensively used today in mathematics, physics, and other sciences. There are two important classes of geometries: Riemannian and nonmetric geometries. In Riemannian world, all of the geometric properties are built from the metric (such as in Euclidean geometry). In nonmetric geometries, on the contrary, distances and angles play no role (examples of this type include affine and projective geometry). In this course, we will give an overview of both of these classes of geometries, following the (algebraic) viewpoint originally suggested by Felix Klein: describing a given geometry using its isometries or symmetry groups. Hence, we will make a systematic use of linear algebra, basic group theory, and also multivariable analysis (especially when discussing curvature, an important invariant appearing in both metric and nonmetric geometries). 

Uren per week  
Onderwijsvorm 
Hoorcollege (LC), Opdracht (ASM), Werkcollege (T)
(Be present at at least 75% of the lectures. Active participation in the tutorials can partially compensate for this and/or result in a higher final grade according to the grading scheme.) 

Toetsvorm 
Opdracht (AST), Schriftelijk tentamen (WE)
(The final grade is obtained as max(E, 0.7E + 0.3H), where E stands for the exam (respectively, the resit) grade and H for the average of the 3 homework assignments. Active participation in the tutorials can improve the homework average H by 10%.) 

Vaksoort  bachelor  
Coördinator  dr. N. Martynchuk  
Docent(en)  dr. N. Martynchuk  
Verplichte literatuur 


Entreevoorwaarden  Knowledge assumed: courses Linear Algebra and (multivariable) Analysis. Additionally, it is assumed that the student is familiar with the basics of Group Theory and Complex Analysis.  
Opmerkingen  The course prepares for the course Analysis on Manifolds and the track Mathematics and Complex Dynamical Systems of the MSc Mathematics.  
Opgenomen in 
