$$\newcommand{\EE}{\mathbb{E}} \newcommand{\HH}{\mathbb{H}} \newcommand{\RR}{\mathbb{R}} \newcommand{\orb}{\mbox{orb}} \newcommand{\Isom}{\operatorname{Isom}} \newcommand{\isom}{\cong} \newcommand{\semi}{\rtimes}$$

## MA243 Geometry Term I 2014-2015

### Schedule

Week Date of Monday Topics Sections in Reid-Szendrői Example sheet Comments
1 Sep. 29 Introduction. Metric spaces and examples. Isometric embeddings and isometries. The Cauchy-Schwarz inequality. Appendix A, 1.1 One See the lovely first chapter of Steele's book for several proofs of the Cauchy-Schwarz inequality.
2 Oct. 6 Platonic solids and $$S^2$$. The triangle inequality in $$\EE^n$$. Distance versus arc-length. Lines, collinearity, parallelity, and Playfair's axiom. Axiomatic systems and models of such. 1.2 - 1.4 Two All five of the Platonic solids can be inscribed in $$S^2$$, after rescaling.
3 Oct. 13 Barycentric coordinates. Isometries preserve barycentric coordinates, are affine maps. Angles. Polarization identity. Isometries preserve angles. Orthonormal bases, orthogonal matrices. Orientation preserving/reversing. Rotations and reflections in $$\EE^2$$. 1.5 - 1.11.1, Appendix B.1 Three
4 Oct. 20 Orthogonal complements and the classification of orthogonal matrices in all dimensions. The parallel postulate and the angle sum of a euclidean triangle. Classification of isometries of $$\EE^2$$. The cosine law. The sine law and its connection to area. 1.11.2 - 1.16.3 (2.1 - 2.6) Four Please read sections 2.1 - 2.6, on composition of isometries.
5 Oct. 27 Spherical distance, great circles, polar duals, and angles. The spherical cosine law and triangle inequality. Classification of isometries of the two-sphere. 3.1 - 3.4 Five
6 Nov. 3 Spherical area, lunes, and Girard's lemma. Hyperbolic trig, Lorentz dot product, the hyperbolic plane. Hyperbolic distance, angles, unit tangent vectors. 3.5 - 3.9 Six
7 Nov. 10 Lorentzian orthogonals, lines parametrized by arclength, characterizing unit tangents. Cosine law and the triangle inequality. Semidirect products, $$\Isom(\EE^2) \isom \RR^2 \semi O(2)$$, $$\Isom(S^2) \isom O(3)$$. 3.10 - 3.11 Seven
8 Nov. 17 $$\Isom(\HH^2) \isom O^+(1,2)$$. Rotations, translations, parabolics. Pencils in the three geometries. Elliptic, parabolic, hyperbolic pencils. Klein model. Independence of the parallel postulate. 3.12 - 3.13 Eight
9 Nov. 24 Area of hyperbolic triangles. Ideal triangles. Affine space, subspaces, transformations. The dimension formula, the affine group. Klein's Erlangen program. Projective equivalence, projective space. 3.14, 4.2 - 4.6, 6.3, 5.2 Nine
10 Dec. 1 Projective line, projective plane. Dimension formula, the projective group. Three-transitivity. Desargues's theorem. 5.3 - 5.11 Ten