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

## MA243 Geometry Term I 2015-2016

### Schedule

Week Date of Monday Topics Sections in Reid-Szendrői Example sheet Comments
1 Oct. 5 Introduction. Metric spaces, examples, nonexamples. Cauchy-Schwarz inequality. Barycentric coordinates. The triangle inequality in $$\EE^n$$. Appendix A, 1.1 One See the lovely first chapter of Steele's book for several proofs of the Cauchy-Schwarz inequality.

There are no support classes this week.

2 Oct. 12 Lines, collinearity, parallelity, barycentric coordinates. Parallel postulate, Playfair's axiom. Axiomatic systems and models of such. Isometric embeddings and isometries. Klein's Erlangen program. Angles. Isometries of $$\EE^n$$ preserve barycentric coordinates. Isometries preserve barycentric coordinates, are affine maps. 1.2 - 1.9 Two
3 Oct. 19 Isometries preserve angles. Polarization identity. Orthonormal bases, orthogonal matrices. Orientation preserving/reversing. Rotations and reflections in $$\EE^2$$. Orthonormal frames, coordinate systems. Orthogonal complements and the classification of orthogonal matrices in all dimensions. 1.10 - 1.13, Appendix B.1 - B.3 Three
4 Oct. 26 Classification of isometries of $$\EE^2$$. The parallel postulate and the angle sum of a euclidean triangle. Cosine, sine. Cosine law, sine law, area. Sphere, great circles, antipodal points. 1.14 - 1.16.3, 2.1 - 2.6, 3.1 Four Questions asked by students.

Sections 2.1 - 2.6 will not be covered in lecture - please read these over the weekend.

We used area to prove the law of sines. However, giving a rigorous definition of area is not easy.

5 Nov. 2 Spherical distance, chordal distance. Spherical angles, spherical triangles. First spherical cosine law and triangle inequality. Polar duals and second cosine law. Isometries of the two-sphere. $$\Phi \from \Orth(3) \to \Isom(S^2)$$ is injective. 3.1 - 3.4 Five Question asked by a student. Note that the six extra-ocular muscles express the three degrees of freedom of an airplane: pitch, yaw, and roll. This corresponds to the fact that $$\SO(3)$$ is three-dimensional.
6 Nov. 9 $$\Phi \from \Orth(3) \to \Isom(S^2)$$ is surjective. Spherical isometries, area, and Girard's lemma. Hyperbolic trig, Lorentz dot product. Space-, light-, and time-like. The hyperbolic plane, the key lemma, hyperbolic distance. Coordinate-free definition of angle. 3.5 - 3.9 Six Questions asked by students.
7 Nov. 16 Lorentzian orthogonals, mixed planes, great hyperbolas. "Unit tangent vectors", lines parametrized by arclength. Cosine law and the triangle inequality. Semidirect products, $$\Isom(\EE^2) \isom \RR^2 \semi \Orth(2)$$, $$\Isom(S^2) \isom \Orth(3)$$. 3.10 - 3.11 Seven Questions asked by students.
8 Nov. 23 $$\Isom(\HH^2) \isom \Orth^+(1,2)$$. Rotations, parabolics, translations. Pencils in the three geometries. Elliptic, parabolic, hyperbolic pencils. Klein model. 3.12 - 3.13 Eight Questions asked by students.
9 Nov. 30 Independence of the parallel postulate. Area of hyperbolic triangles. Ideal triangles. Affine space, affine subspaces, transformations, the affine group. The dimension formula, Klein's Erlangen program. 3.14, 4.2 - 4.6, 6.3 Nine
10 Dec. 7 Projective equivalence, projective space. Projective lines, the projective plane. Dimension formula, the projective group. Fundamental theorem of projective geometry. Three-transitivity, cross-ratios, distance in Klein model. 5.2 - 5.8 Ten

Eleven
Questions asked by students.