There are several terms which are included in the title of this book,
such as "Lie groups", "geometry", and "homogeneous spaces", so it
maybe worthwhile to provide an explanation about their relation-
ships. We will start with the term "geometry", which most readers
are familiar with.
Geometry comes from the Greek word "^eoojieTpeLv", which means
to measure land. Various techniques for this purpose, including other
practical calculations, were developed by the Babylonians, Egyptians,
and Indians. Beginning around 500 BC, an amazing development was
accomplished, whereby Greek thinkers abstracted a set of definitions,
postulates, and axioms from the existing geometric knowledge, and
showed that the rest of the entire body of geometry could be de-
duced from these. This process led to the creation of the book by
Euclid entitled The Elements. This is what we refer to as Euclidean
However, the fifth postulate of Euclid (the parallel postulate)
attracted the attention of several mathematicians, basically because
there was a feeling that it would be possible to prove it by using
the first four postulates. As a result of this, new geometries ap-
peared (elliptic, hyperbolic), in the sense that they are consistent
without using Euclid's fifth postulate. These geometries are known
as Non-Euclidean Geometries, and some of the mathematicians that
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