Introduction
xv
Chapter 1 starts with a simple example of a Lie group that ex-
hibits the manifold and group structure. Then we give a brief review
of manifolds, and then we proceed with the definition of a Lie group.
We define the Lie algebra of a Lie group as the tangent space at
the identity element of the group, and alternatively as the set of its
one-parameter subgroups. We also list a simplified version of Lie's
theorems.
In Chapter 2, after discussing a few elementary concepts about
representations, we develop the appropriate tools that are needed for
the classification of the compact and connected Lie groups. These are
the adjoint representation, and the maximal torus of a Lie group. We
also introduce a very useful tool, the Killing form, and we provide a
brief insight through the complex semisimple Lie algebras.
Chapter 3 starts with a brief review of Riemannian manifolds,
and then discusses a way to make a Lie group into a Riemannian
manifold. The metrics which are important here are the bi-invariant
metrics, and with respect to such metrics we give formulas for the
connection and the various types of curvatures.
In Chapter 4 we define the notion of a homogeneous space and
provide several examples. We discuss the reductive homogeneous
spaces, and the isotropy representation of such a space.
The geometry of a homogeneous space is discussed in Chapter 5,
where we show how a homogeneous space G/K can become a Rie-
mannian manifold (so we obtain a Riemannian homogeneous space).
The important metrics here are the G-invariant metrics. Formulas
are presented for the connection and the various types of curvatures.
In Chapters 6 and 7 we discuss two important, and generally non-
overlapping, classes of homogeneous spaces, which are the symmetric
spaces and the generalized flag manifolds. One of the most significant
advances of the twentieth century mathematics is Cartan's classifica-
tion of semisimple Lie groups. This leads to the classification of these
two classes of homogeneous spaces. These spaces have many appli-
cations in real and complex analysis, topology, geometry, dynamical
systems, and physics.
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