Week 8
Space-like metric geometry:
Hyperbolic geometry
This week, we turn to the second standard example of non-Euclidean geometry,
namely to hyperbolic geometry. It will not be a topic of this module, but
one can motivate our model of hyperbolic geometry by observing that it
naturally occurs in the underlying geometry of general relativity, namely
as a sphere with time-like radius.
Similarly to the discussion of
plane and spherical geometry, I start by fixing the
notions of hyperbolic distance and lines, and by
showing that in hyperbolic geometry, collinearity is a metric property.
Next, we will study hyperbolic angles and hyperbolic trigonometry as
well as hyperbolic motions.
Finally, investigating the angular defect and ideal triangles in
hyperbolic geometry, we will find some genuine differences between hyperbolic
geometry and the other geometries this module has covered so far.
Helpful literature:
-
M. Reid, Geometry and Topology, chapters of forthcoming
textbook by M. Reid and B. Szendröi, available from General Office:
Sections 4.7-4.13.