MATH3201/4141 Geometry III/IV
The course deals with the relationship between group theory and geometry as outlined by Felix Klein in the late 19th century in his famous "Erlanger Program". This will be discussed through different types of geometries in 2-dimensions, namely Euclidean, affine, projective, Möbius and hyperbolic geometry.
Each of these geometries may be characterized as the study of properties of figures which are preserved by a certain group of transformations. For example, Euclidean geometry of the plane is concerned with properties which depend on the notion of distance. The corresponding group of distance preserving transformations is the Euclidean group consisting of translations, rotations, reflections and glide reflections. The affine group, which contains the Euclidean group, is the group of transformations of the plane which send line to lines and the corresponding geometry, which is more general than Euclidean geometry, is called affine geometry. Projective geometry, which is a further generalization, provides a powerful tool for describing many aspects of the geometry of the plane particularly those concerned with the ellipse, parabola and hyperbola. Möbius geometry is the geometry of lines and circles in the plane and the Möbius group is the group of Möbius transformations of the extended complex plane familiar from complex analysis.
All the classical geometries mentioned above play an important role in the construction of hyperbolic geometry. For more than two thousand years, geometers tried to decide whether or not Euclid's parallel postulate is a consequence of his other four postulates. In 1830, hyperbolic geometry was developed independently by Bolyai, Gauss and Lobatchevsky, and so settled this problem. Since its discovery, the hyperbolic geometry has had a profound effect on the foundations of geometry, recent development of topology, and on various aspects of mathematical physics, especially relativity theory.
Outline of Course
Aim: To give students a basic grounding in various aspects of plane geometry. In particular, to elucidate different types of plane geometries and to show how these may be handled from a group theoretic viewpoint.
- Euclidean Geometry: Basics of Euclidean geometry. Isometries, their linear part and group of isometries, reflections as generators of the group, fixed points. Conjugacy classes. Discrete subgroups of the isometry group. Fundamental domains. Torus and Klein bottle as quotients of ℝ² by discrete subgroups which act freely. Klein's Erlanger Program and the relationship between group theory and geometry.
- Spherical geometry: Spherical lines, triangle inequality, polars. Congruence of spherical triangles, area of spherical triangles, law of sines and law of cosines. SO(3) as the group of orientation preserving isometries, reflections as generators of the whole isometry group.
- Affine and projective geometries: Projective line and projective plane, cross- ratio, homogeneous coordinates, projective transformations as compositions of projections from a point. Affine subgroup of the projective group.
- Hyperbolic geometry: Model in two-sheet hyperboloid, PO(2,1) as the isometry group, reflections as generators. Hyperbolic lines, half-planes. Projection to Poincaré disc.
- Möbius Transformations: Möbius transformations. Triple transitivity. Conformal transformations are Möbius. Conjugacy classes. Inversion, the inversive group. Similarities of ℝ² and isometries of S² as elements of the inversive group. Circle-preserving transformations of S² are inversive. Orthogonal circles. Cross-ratio. Characterisation of a circle through three points in terms of cross-ratio.
- Poincaré models: Upper half-plane: hyperbolic lines, PSL(2;ℝ) as the group of orientation preserving isometries. Poincaré disc: lines, angles, PU(1,1) as the group of orientation preserving isometries.
- Elementary hyperbolic geometry: Euclid's axioms, parallel postulate, parallel lines in the hyperbolic plane, angle of parallelism. Hyperbolic triangles, sum of angles, congruence of triangles, hyperbolic trigonometry, law of sines, law of cosines, area of hyperbolic triangles.
- Isometries: classification, conjugacy classes, invariant sets (circles, horocycles and equidistant curves).
- Klein disc model: lines, angles, horocycles.
- Reflection groups: reflection groups on the sphere, Euclidean plane and hyperbolic plane: fundamental domains, Gram matrices, classification.
- Further topics: Hyperbolic surfaces: brief treatment (by example) of surfaces of genus greater than 1 as quotients of the hyperbolic plane by discrete subgroups which act freely, pants decompositions. Hyperbolic space.