1. Euclidean, Projective and Lobachevsky
Geometries: a historical summary.
The projective plane and the
Riemann sphere. The cross-ratio.
2. The models of a hyperbolic plane, the lines. A
summary on Metrics and Riemannian metrics.
3. Length, angles, geodesics and isometries in
Euclidean, Elliptic and Hyperbolic Geometries.
4. Group PSL(2,Z)
and its action. M\"obius transformations as isometries
of H.
5. The fixed points and asymptotics of M\"obius transformations. Conjugacy and the trace. Elliptic and hyperbolic transformations.
6. Hyperbolic triangles, conformality. Gaussian curvature and the Gauss-Bonnet theorem.
7. Hyperbolic Trigonometry. The hyperboloid model and
the Kleinian (projective) model.
8. The Minkovsky space and Lorentz
transformations. Relation to the Special Relativity Theory.
9. Surfaces: polygonal models and tilings of H.
10-12. Fuchsian groups and
fundamental domains.