Non-Euclidean geometry

Despite the fact that non-Euclidean geometry has found its use in numerous applications (the most striking example being 3-dimensional topology), it has retained a kind of exotic and romantic element. This course intends to be a businesslike introduction to non-Euclidean geometry for nonexperts.


Curriculum:

  1. Prerequisites from linear algebra, point set topology, group theory, metric spaces.
  2. Axioms for plane geometry.
  3. The inversive models.
  4. The hyperboloid and the Klein model.
  5. The geometry of the sphere.
  6. Some computations in the hyperbolic plane and on the sphere.
  7. Hyperbolic isometries.
  8. Convex polygons.
  9. Isoperimetric inequality in non-Euclidean geometry.
  10. Hyperbolic surfaces.


Textbooks

  • I. S. Iversen, Hyperbolic geometry, Cambridge Univ. Press 1993.
  • H. S. Coxeter, Non-Euclidean Geometry, Toronto Univ. Press, 1957.