Mathematics BSc
Course description
2013.
Borromean Rings
Geometry 2
Hours
lect+pc		 Credits
lect+pc		 Assessment Specialization Course code
lect/pc		 Semester Status
2 + 2 2 + 3 exam +
term grade		 pure math. mm1c1ge3m
mm1c2ge3m		 3 compulsory
Strong Weak Prerequisites
Practice class
Strong:
Geometry1L (mm1c1ge2)
Strong:
Analysis2L (mm1c1an2) or
Foundations of analysisL (mm1c1ap2)
Lecture
Weak:
practice class
Weak:
Introduction to topologyL-p (mm1c1to3m)
Literature
  • Marcel Berger: Geometry I. Springer, 1987.
Syllabus
  • Euclidean isometries. The concept of Euclidean space and Euclidean isometry. Translations and orthogonal transformations. Natural decomposition of isometries. Classification of isometries in dimensions 2 and 3. Orthogonal decompositions, projections, symmetries, and reflections. Representing isometries as products of reflections.
  • Orthogonal groups: Topological and algebraic properties of orthogonal groups. Simplicity of SO(3). Geometry of quaternions.
  • Regular polytopes: Symmetry groups of regular polygons and polyhedra as finite subgroups of SO(2) and SO(3). Definition and constructions of regular polytopes. Classification of regular polytopes in higher dimensions.
  • Similarities: Similarity transformations in Euclidean space. The group of similarities. Sphere preserving maps.
  • Inversive geometry: Power of points with regard to a hypersphere, the power hyperplane. Inversion in a hypersphere. Inverses of affine subspaces and spheres, conformality. Stereographic projection. Inversive space, Möbius transformations, Möbius group. The Poincaré extension of Möbius transformations.
  • Volume and surface area: The concept of volume in Euclidean space. Approximation of convex bodies with polytopes. Surface area of polytopes and convex bodies. Volume and surface area of balls. The Hausdorff metric. Volume and surface area as continuous functions. The Blaschke selection theorem. The Steiner-Minkowski theorem. Steiner symmetrization. Isodiametric and isoperimetric inequalities.
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