Random Processes in Hyperbolic Spaces Hyperbolic Brownian ...

Introduction
Euclid wrote his famous Elements around 300 B.C. In the first book of the Elements Euclid develops
plane geometry starting with the following five postulates
• A straight line may be drawn from any point to any other point.
• A finite straight line may be extended continuously in a straight line.
• A circle may be drawn with any center and any radius.
• All right angles are equal.
• If a straight line falling on two straight lines makes the interior angles on the same side less
than two right angles, the two straight lines, if extended indefinitely, meet on the side on
which the angles are less than two right angles.
Euclid’s fifth postulate is equivalent to the modern postulate of Euclidean geometry
• Through a point outside a given infinite line there is one and only one infinite straight line
parallel to the given line.
For two thousand years it was believed that the fifth postulate could be derived from the other
four postulates and for centuries the mathematicians attempted to prove the fifth postulate. It
was not until the nineteenth century that the fifth postulate was finally shown to be independent.
The proof of this independence was the result of a completely unexpected discovery: the denial of
the fifth postulate leads to a new consistent geometry.
It was Gauss who first made this discovery. It was known from the eighteenth century that the
fifth postulate is equivalent to the theorem that the sum of the angles of a triangle is 180 o , Gauss,
assuming that the sum of the angles of a triangle is less than 180 o , discovered the non-Euclidean
geometry.
Only few years later Lobachevsky published the first account of non-Euclidean geometry in 1829
in a paper entitled On the principle of geometry. In 1832, Bolay published an independent account
of non-Euclidean geometry in a paper entitled The absolute science of space.
Today the non-Euclidean geometry of Gauss, Lobachevsky and Bolay is called hyperbolic geometry,
and the term non-Euclidean geometry refers to any geometry that is not Euclidean.
After defining spaces of variable curvature and noting that the Euclidean space is a space of
zero curvature, Riemann considered spaces of constant non zero curvature. An example of a ndimensional
Riemannian space of constant positive curvature is the n-dimensional sphere in the