any other point of G/K by using certain maps (translations). Hence,
all calculations reduce to a single point which, for simplicity, can be
chosen to be the identity coset o = eK £ G/K. Furthermore, in
an important special case where the homogeneous space is reductive,
then the tangent space of G/K at o can be identified in a natural way
with a subspace of g.
As a consequence of this, many hard problems in homogeneous
geometry can be formulated in terms of the group G and the subgroup
K, and then in terms of their corresponding infinitesimal objects g
and £. Such an infinitesimal approach enables us to use linear alge-
bra to tackle non-linear problems (from geometry, analysis, or theory
of differential equations). For example, the equations satisfied by
an Einstein metric (these, according to general relativity, describe
the evolution of the universe) are a complicated non-linear system
of partial differential equations. However, for G-invariant metrics on
a homogeneous space, this system reduces to a system of algebraic
equations, which can be solved in many cases.
There is a large variety of applications of Lie groups in mathe-
matics. They appear in various ways beyond differential geometry,
such as algebraic topology, harmonic analysis, and differential equa-
tions, to name a few. They also possess important applications in
physics, since they become involved in field theories in many ways.
In fact, certain classical Lie groups appear as the building blocks in
various physical theories of matter. Homogeneous spaces, in turn,
have been employed in the physics of elementary particles as mod-
els called supersymmetric sigma models. Also, what physicists call
coherent states, are in one-to-one correspondence with elements in a
Before we proceed to the description of the chapters of this book,
we would like to mention that the two generalizations of Euclidean
geometry that we mentioned, namely that of Riemann and that of
Klein, were unified by E. Cartan in his theory of espaces generalizes.
In Cartan's geometry, at each point m of M, there is a Klein-style
geometry in the tangent space. That is to say, Cartan took Klein's
geometry and made it local to each tangent space.